{
"query": {
"display": "$$y^{^{\\prime\\prime}}\\left(x\\right)-3y^{^{\\prime}}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}$$",
"symbolab_question": "ODE#y^{\\prime \\prime }(x)-3y^{\\prime }(x)+2y=\\frac{1}{1+e^{-x}}"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "ODE",
"subTopic": "ConstCoeffLinearNonHomogeneous",
"default": "y=c_{1}e^{2x}+c_{2}e^{x}+(-1-e^{-x}+\\ln(1+e^{-x}))e^{2x}+\\ln(1+e^{-x})e^{x}",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}:{\\quad}y=c_{1}e^{2x}+c_{2}e^{x}+\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}$$",
"input": "y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}",
"steps": [
{
"type": "interim",
"title": "Solve linear ODE:$${\\quad}y=c_{1}e^{2x}+c_{2}e^{x}+\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}$$",
"input": "y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}",
"steps": [
{
"type": "definition",
"title": "Second order linear non-homogeneous differential equation with constant coefficients",
"text": "A second order linear, non-homogeneous ODE has the form of $$ay''+by'+cy=g\\left(x\\right)$$"
},
{
"type": "step",
"primary": "The general solution to $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=g\\left(x\\right)$$ can be written as<br/>$$y=y_h+y_p$$<br/>$$y_h$$ is the solution to the homogeneous ODE $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=0$$<br/>$$y_p$$, the particular solution, is any function that satisfies the non-homogeneous equation "
},
{
"type": "interim",
"title": "Find $$y_h$$ by solving $$y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=0:{\\quad}y=c_{1}e^{2x}+c_{2}e^{x}$$",
"input": "y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=0",
"steps": [
{
"type": "definition",
"title": "Second order linear homogeneous differential equation with constant coefficients",
"text": "A second order linear, homogeneous ODE has the form of $$ay''+by'+cy=0$$"
},
{
"type": "step",
"primary": "For an equation $$ay''+by'+cy=0$$, assume a solution of the form $$e^{γx}$$",
"secondary": [
"Rewrite the equation with $$y=e^{γx}$$"
],
"result": "\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime\\prime}}-3\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime}}+2e^{γx}=0"
},
{
"type": "interim",
"title": "Simplify $$\\left(\\left(e^{γx}\\right)\\right)^{\\prime\\prime}-3\\left(\\left(e^{γx}\\right)\\right)^{\\prime}+2e^{γx}=0:{\\quad}e^{γx}\\left(γ^{2}-3γ+2\\right)=0$$",
"steps": [
{
"type": "step",
"result": "\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime\\prime}}-3\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime}}+2e^{γx}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γx}\\right)^{\\prime\\prime}=γ^{2}e^{γx}$$",
"input": "\\left(e^{γx}\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(e^{γx}\\right)^{\\prime}=e^{γx}γ$$",
"input": "\\left(e^{γx}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γx}\\left(γx\\right)^{\\prime}$$",
"input": "\\left(e^{γx}\\right)^{\\prime}",
"result": "=e^{γx}\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γx$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γx\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γx\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γx$$",
"result": "=e^{γx}\\left(γx\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcBDY8sG8wkKf3KVMoxpWVQssjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5M1thhQ/Ed/UbgOWE3OQBzQDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γx\\right)^{\\prime}=γ$$",
"input": "\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γx^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EVB+jU4LMswLFUscsVdfVSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tV6EkZL7Rz6t72e1SEoLgfPvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γx}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(e^{γx}γ\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(e^{γx}γ\\right)^{\\prime}=γ^{2}e^{γx}$$",
"input": "\\left(e^{γx}γ\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γ\\left(e^{γx}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γx}\\left(γx\\right)^{\\prime}$$",
"input": "\\left(e^{γx}\\right)^{\\prime}",
"result": "=e^{γx}\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γx$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γx\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γx\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γx$$",
"result": "=e^{γx}\\left(γx\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcBDY8sG8wkKf3KVMoxpWVQssjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5M1thhQ/Ed/UbgOWE3OQBzQDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γx\\right)^{\\prime}=γ$$",
"input": "\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γx^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EVB+jU4LMswLFUscsVdfVSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tV6EkZL7Rz6t72e1SEoLgfPvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=γe^{γx}γ"
},
{
"type": "interim",
"title": "Simplify $$γe^{γx}γ:{\\quad}γ^{2}e^{γx}$$",
"input": "γe^{γx}γ",
"result": "=γ^{2}e^{γx}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$γγ=\\:γ^{1+1}$$"
],
"result": "=e^{γx}γ^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=e^{γx}γ^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ogc+QG1cya0zQe2NAtljy96GQqufR6tr2vPxOUv7H++vcvW4o70lTWJV7TqReTsbmx4rCXhbsN+br+uOYP22UU3kCh3oevUunZ7/b0qFKBStCRMtul5SOs/SBwPTbaWuzHQDH6pWoNP3G07HzBhaLA=="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=γ^{2}e^{γx}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γx}-3\\left(e^{γx}\\right)^{^{\\prime}}+2e^{γx}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γx}\\right)^{\\prime}=e^{γx}γ$$",
"input": "\\left(e^{γx}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γx}\\left(γx\\right)^{\\prime}$$",
"input": "\\left(e^{γx}\\right)^{\\prime}",
"result": "=e^{γx}\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γx$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γx\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γx\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γx$$",
"result": "=e^{γx}\\left(γx\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcBDY8sG8wkKf3KVMoxpWVQssjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5M1thhQ/Ed/UbgOWE3OQBzQDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γx\\right)^{\\prime}=γ$$",
"input": "\\left(γx\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γx^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EVB+jU4LMswLFUscsVdfVSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tV6EkZL7Rz6t72e1SEoLgfPvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γx}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γx}-3e^{γx}γ+2e^{γx}=0"
},
{
"type": "step",
"primary": "Factor $$e^{γx}$$",
"result": "e^{γx}\\left(γ^{2}-3γ+2\\right)=0"
}
],
"meta": {
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "step",
"result": "e^{γx}\\left(γ^{2}-3γ+2\\right)=0"
},
{
"type": "interim",
"title": "Solve $$e^{γx}\\left(γ^{2}-3γ+2\\right)=0:{\\quad}γ=2,\\:γ=1$$",
"input": "e^{γx}\\left(γ^{2}-3γ+2\\right)=0",
"steps": [
{
"type": "step",
"primary": "Since $$e^{γx}\\ne\\:0$$, solving $$e^{γx}\\left(γ^{2}-3γ+2\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}-3γ+2=0$$",
"result": "γ^{2}-3γ+2=0"
},
{
"type": "interim",
"title": "Solve with the quadratic formula",
"input": "γ^{2}-3γ+2=0",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}",
"steps": [
{
"type": "definition",
"title": "Quadratic Equation Formula:",
"text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$"
},
{
"type": "step",
"primary": "For $${\\quad}a=1,\\:b=-3,\\:c=2$$",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}"
}
],
"meta": {
"interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "$$\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}=1$$",
"input": "\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:1}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-3\\right)^{2}=3^{2}$$"
],
"result": "=\\sqrt{3^{2}-4\\cdot\\:1\\cdot\\:2}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:2=8$$",
"result": "=\\sqrt{3^{2}-8}"
},
{
"type": "step",
"primary": "$$3^{2}=9$$",
"result": "=\\sqrt{9-8}"
},
{
"type": "step",
"primary": "Subtract the numbers: $$9-8=1$$",
"result": "=\\sqrt{1}"
},
{
"type": "step",
"primary": "Apply rule $$\\sqrt{1}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77dtxLd9UeyppCvogsL4v83GW/jTwUrqHAcqr1cHDsBBw4ba/3KCEGXsM04IlGc/HzMFYmi1F5Hg/ibpEToVnY247KVn/9CN730bZvRQSpulUlJGhz/9/49m9VyFXtj5ecZb+NPBSuocByqvVwcOwEEDVEq/Lyx8C1I/8Q6ctddo="
}
},
{
"type": "step",
"primary": "Separate the solutions",
"result": "{γ}_{1}=\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1},\\:{γ}_{2}=\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}"
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1}:{\\quad}2$$",
"input": "\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\frac{3+1}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Add the numbers: $$3+1=4$$",
"result": "=\\frac{4}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=\\frac{4}{2}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{4}{2}=2$$",
"result": "=2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BRRger6gkjQuZFNUJO1CQrJqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUDMZsebS69oihnBNBSMiS5tayBhLPCAZUhx1M1IbRaw+d7DaIenKLTiwZv2anSwwjQ=="
}
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}:{\\quad}1$$",
"input": "\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\frac{3-1}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Subtract the numbers: $$3-1=2$$",
"result": "=\\frac{2}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=\\frac{2}{2}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{a}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fb8/QwpiZ9fKB0o6NPD4MrJqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiK7J5E5gGi2xwchkRMjoVJ7jMZsebS69oihnBNBSMiS5sFZg/Lik3tDW+uYEraYwryd7DaIenKLTiwZv2anSwwjQ=="
}
},
{
"type": "step",
"primary": "The solutions to the quadratic equation are:",
"result": "γ=2,\\:γ=1"
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "γ=2,\\:γ=1"
},
{
"type": "step",
"primary": "For two real roots $$γ_{1}\\ne\\:γ_{2}$$, the general solution takes the form:$${\\quad}y=c_{1}e^{γ_{1}\\:x}+c_{2}e^{γ_{2}\\:x}$$",
"result": "y=c_{1}e^{2x}+c_{2}e^{x}"
}
],
"meta": {
"solvingClass": "ODE",
"interimType": "Generic Find By Solving Title 2Eq"
}
},
{
"type": "interim",
"title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}:{\\quad}y_{p}=\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}$$",
"steps": [
{
"type": "step",
"primary": "For the non-homogeneous part $$g\\left(x\\right)=\\frac{1}{1+e^{-x}}$$, assume a solution of the form:<br/>$$y_{p}=u_{1}y_{1}+u_{2}y_{2}$$<br/>Where $$y_{1}$$ and $$y_{2}$$ are solutions of homogeneous equation $$y_{h}=c_{1}y_{1}+c_{2}y_{2}$$, <br/> $${u_{1}}$$ and $${u_{2}}$$ are solutions to the system of equations:<br/>$$\\begin{pmatrix}\\left(u_{1}\\right)^{^{\\prime}}y_{1}+\\left(u_{2}\\right)^{^{\\prime}}y_{2}=0\\\\\\left(u_{1}\\right)^{^{\\prime}}\\left(y_{1}\\right)^{^{\\prime}}+\\left(u_{2}\\right)^{^{\\prime}}\\left(y_{2}\\right)^{^{\\prime}}=g\\left(x\\right)\\end{pmatrix}$$<br/>Which implies",
"secondary": [
"$$u_{1}=\\int\\:-\\frac{y_{2}g\\left(x\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dx$$",
"$$u_{2}=\\int\\:\\frac{y_{1}g\\left(x\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dx$$",
"Where Wronskian $$W\\left(y_{1},\\:y_{2}\\right)=y_{1}\\left(y_{2}\\right)^{^{\\prime}}-\\left(y_{1}\\right)^{^{\\prime}}y_{2}$$"
]
},
{
"type": "step",
"primary": "Homogeneous solutions:",
"secondary": [
"$$y_{1}=e^{2x}$$",
"$$y_{2}=e^{x}$$"
]
},
{
"type": "interim",
"title": "$$\\left(y_{1}\\right)^{^{\\prime}}:{\\quad}e^{2x}\\cdot\\:2$$",
"input": "\\left(e^{2x}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{2x}\\left(2x\\right)^{\\prime}$$",
"input": "\\left(e^{2x}\\right)^{\\prime}",
"result": "=e^{2x}\\left(2x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=2x$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(2x\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(2x\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=2x$$",
"result": "=e^{2x}\\left(2x\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71GjX8/WOlVQoR3720Yfm9aRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOcTTG1MQjF4JOCdGL5fDxZfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "interim",
"title": "$$\\left(2x\\right)^{\\prime}=2$$",
"input": "\\left(2x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2x^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=2\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=2",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iEluvAhAB6qVseHmXOoGm8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMolkIAcH8DCZe9Bzo2oZbtQQDuUeqnBwgbzjs2dJUq2K"
}
},
{
"type": "step",
"result": "=e^{2x}\\cdot\\:2"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(y_{2}\\right)^{^{\\prime}}:{\\quad}e^{x}$$",
"input": "\\left(e^{x}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{x}\\right)^{\\prime}=e^{x}$$",
"result": "=e^{x}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBKEUujzBr/OwXKSSv8Lr7msNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqoYkL0LptyesbbdTForD1TM+mtSooHHgz8yj8VjvtwX+TK+f0kTaFpliuy3OWH0/pl"
}
},
{
"type": "step",
"primary": "$$W\\left(y_{1},\\:y_{2}\\right)=y_{1}\\left(y_{2}\\right)^{^{\\prime}}-\\left(y_{1}\\right)^{^{\\prime}}y_{2}$$",
"result": "W\\left(y_{1},\\:y_{2}\\right)=e^{2x}e^{x}-e^{2x}\\cdot\\:2e^{x}"
},
{
"type": "step",
"result": "W\\left(y_{1},\\:y_{2}\\right)=-e^{3x}"
},
{
"type": "step",
"primary": "$$u_{1}=\\int\\:-\\frac{y_{2}g\\left(x\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dx$$",
"result": "u_{1}=\\int\\:-\\frac{e^{x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx"
},
{
"type": "interim",
"title": "$$\\int\\:-\\frac{e^{x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx=-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)+C$$",
"input": "\\int\\:-\\frac{e^{x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx",
"steps": [
{
"type": "interim",
"title": "Simplify $$-\\frac{e^{x}\\frac{1}{1+e^{-x}}}{-e^{3x}}:{\\quad}\\frac{1}{e^{2x}\\left(1+e^{-x}\\right)}$$",
"input": "-\\frac{e^{x}\\frac{1}{1+e^{-x}}}{-e^{3x}}",
"steps": [
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$",
"result": "=-\\left(-\\frac{e^{x}\\frac{1}{e^{-x}+1}}{e^{3x}}\\right)"
},
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\frac{e^{x}\\frac{1}{1+e^{-x}}}{e^{3x}}"
},
{
"type": "interim",
"title": "Multiply $$e^{x}\\frac{1}{1+e^{-x}}\\::{\\quad}\\frac{e^{x}}{1+e^{-x}}$$",
"input": "e^{x}\\frac{1}{1+e^{-x}}",
"result": "=\\frac{\\frac{e^{x}}{1+e^{-x}}}{e^{3x}}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:e^{x}}{1+e^{-x}}"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:e^{x}=e^{x}$$",
"result": "=\\frac{e^{x}}{1+e^{-x}}"
}
],
"meta": {
"interimType": "Generic Multiply Title 1Eq"
}
},
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$",
"result": "=\\frac{e^{x}}{\\left(1+e^{-x}\\right)e^{3x}}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}\\:=\\:x^{a-b}$$",
"secondary": [
"$$\\frac{e^{x}}{e^{3x}}=e^{x-3x}=e^{-2x}$$"
],
"result": "=\\frac{1}{e^{2x}\\left(e^{-x}+1\\right)}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:\\frac{1}{e^{2x}\\left(1+e^{-x}\\right)}dx"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$",
"secondary": [
"$$\\frac{1}{e^{2x}\\left(1+e^{-x}\\right)}=e^{-2x}\\frac{1}{\\left(1+e^{-x}\\right)}$$"
],
"result": "=\\int\\:e^{-2x}\\frac{1}{1+e^{-x}}dx",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "Simplify $$e^{-2x}\\frac{1}{1+e^{-x}}:{\\quad}\\frac{e^{-2x}}{1+e^{-x}}$$",
"input": "e^{-2x}\\frac{1}{1+e^{-x}}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:e^{-2x}}{1+e^{-x}}"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:e^{-2x}=e^{-2x}$$",
"result": "=\\frac{e^{-2x}}{1+e^{-x}}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:\\frac{e^{-2x}}{1+e^{-x}}dx"
},
{
"type": "interim",
"title": "Apply u-substitution",
"input": "\\int\\:\\frac{e^{-2x}}{1+e^{-x}}dx",
"steps": [
{
"type": "definition",
"title": "Integral Substitution definition",
"text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$",
"secondary": [
"Substitute: $$u=1+e^{-x}$$"
]
},
{
"type": "interim",
"title": "$$\\frac{du}{dx}=-e^{-x}$$",
"input": "\\left(1+e^{-x}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=1^{^{\\prime}}+\\left(e^{-x}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$1^{\\prime}=0$$",
"input": "1^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/5pHl7jX6nP6oSkaDnJ8dqboRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhNUy0PGUeVurnmhVFT+4jBc="
}
},
{
"type": "interim",
"title": "$$\\left(e^{-x}\\right)^{\\prime}=-e^{-x}$$",
"input": "\\left(e^{-x}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{-x}\\left(-x\\right)^{\\prime}$$",
"input": "\\left(e^{-x}\\right)^{\\prime}",
"result": "=e^{-x}\\left(-x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=-x$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(-x\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(-x\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=-x$$",
"result": "=e^{-x}\\left(-x\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GSIQlKG9310R1ElGl3CVfaRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOB6LdUC4vDg25Wev+HuCzhfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "interim",
"title": "$$\\left(-x\\right)^{\\prime}=-1$$",
"input": "\\left(-x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=-x^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=-1"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UM4LDHFZjLzDjY8O/P1Ab8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/g69GyryCsZzNNAEljQyUl0eplVd2gGs/MYc2HGVEvpw+gc+cK9cns8BOox/q2OecQ"
}
},
{
"type": "step",
"result": "=e^{-x}\\left(-1\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-e^{-x}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=0-e^{-x}"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-e^{-x}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:du=-e^{-x}dx$$"
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:dx=\\left(-e^{x}\\right)du$$"
},
{
"type": "step",
"result": "=\\int\\:\\frac{e^{-2x}}{u}\\left(-e^{x}\\right)du"
},
{
"type": "interim",
"title": "Simplify $$\\frac{e^{-2x}}{u}\\left(-e^{x}\\right):{\\quad}-\\frac{e^{-x}}{u}$$",
"input": "\\frac{e^{-2x}}{u}\\left(-e^{x}\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{e^{-2x}}{u}e^{x}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=-\\frac{e^{-2x}e^{x}}{u}"
},
{
"type": "interim",
"title": "$$e^{-2x}e^{x}=e^{-x}$$",
"input": "e^{-2x}e^{x}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$e^{-2x}e^{x}=\\:e^{-2x+x}$$"
],
"result": "=e^{-2x+x}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add similar elements: $$-2x+x=-x$$",
"result": "=e^{-x}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FowRfh810+9bpnHz+cS5glXTSum/z5kLpMzXS1UJIex9upyvpSJ+axyVrxsRkkAK/z//r+dXk7h9vxeDCLuZqpKYea5wkxBOE0qTl4Xr3ldhJ8W6EF+Iw3rbaWc1uD4S"
}
},
{
"type": "step",
"result": "=-\\frac{e^{-x}}{u}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{e^{-x}}{u}du"
},
{
"type": "interim",
"title": "$$u=1+e^{-x}\\quad\\Rightarrow\\quad\\:e^{-x}=u-1$$",
"input": "1+e^{-x}=u",
"steps": [
{
"type": "interim",
"title": "Move $$1\\:$$to the right side",
"input": "1+e^{-x}=u",
"result": "e^{-x}=u-1",
"steps": [
{
"type": "step",
"primary": "Subtract $$1$$ from both sides",
"result": "1+e^{-x}-1=u-1"
},
{
"type": "step",
"primary": "Simplify",
"result": "e^{-x}=u-1"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Equations"
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{u-1}{u}du"
}
],
"meta": {
"interimType": "Integral U Substitution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70hb/s63F1m65pyLf6NpX6R8JURM0P4tpPzEvj8+iLpfLI71+ylVDvwHghUiHeEt93Gam9ro5jdhRGQuTP5rGCB4l9+XUP4dNZSL8WpmbbeT4j27KJ23AvytEJKx61QAnz9psU5ldmEOJUKwbBRrOQKqkIX6kseCKdEth+cILnwy7iOSVUKCvHweh2lT4eXSw8TaqbJFwAzvwA3RhTAPNgI="
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{u-1}{u}du"
},
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=-\\int\\:\\frac{u-1}{u}du"
},
{
"type": "interim",
"title": "Expand $$\\frac{u-1}{u}:{\\quad}1-\\frac{1}{u}$$",
"input": "\\frac{u-1}{u}",
"steps": [
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$",
"secondary": [
"$$\\frac{u-1}{u}=\\frac{u}{u}-\\frac{1}{u}$$"
],
"result": "=\\frac{u}{u}-\\frac{1}{u}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{a}=1$$",
"secondary": [
"$$\\frac{u}{u}=1$$"
],
"result": "=1-\\frac{1}{u}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Expand Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tmJ+4RiVlGO9IgEmgGy2pQCWKUbvV6WK3fDUgFtg3Q9Q7GQHzM2DoFG/VzJXJyOZQXlgPFRFcFhBuY7t6rjpnIEFMST8lDZxn1Yq5HMKVTvDpCb0LomfCc/BDlzuuIuGIbxXdJTNg73LMCcLa8vk0Q=="
}
},
{
"type": "step",
"result": "=-\\int\\:1-\\frac{1}{u}du"
},
{
"type": "step",
"primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$",
"result": "=-\\left(\\int\\:1du-\\int\\:\\frac{1}{u}du\\right)"
},
{
"type": "interim",
"title": "$$\\int\\:1du=u$$",
"input": "\\int\\:1du",
"steps": [
{
"type": "step",
"primary": "Integral of a constant: $$\\int{a}dx=ax$$",
"result": "=1\\cdot\\:u"
},
{
"type": "step",
"primary": "Simplify",
"result": "=u",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right)$$",
"input": "\\int\\:\\frac{1}{u}du",
"steps": [
{
"type": "step",
"primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right),\\:$$assuming a complex-valued logarithm",
"result": "=\\ln\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "step",
"result": "=-\\left(u-\\ln\\left(u\\right)\\right)"
},
{
"type": "step",
"primary": "Substitute back $$u=1+e^{-x}$$",
"result": "=-\\left(1+e^{-x}-\\ln\\left(1+e^{-x}\\right)\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)",
"meta": {
"solvingClass": "Solver"
}
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)+C",
"meta": {
"title": {
"extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "step",
"result": "u_{1}=-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)"
},
{
"type": "step",
"primary": "$$u_{2}=\\int\\:\\frac{y_{1}g\\left(x\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dx$$",
"result": "u_{2}=\\int\\:\\frac{e^{2x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx"
},
{
"type": "interim",
"title": "$$\\int\\:\\frac{e^{2x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx=\\ln\\left(1+e^{-x}\\right)+C$$",
"input": "\\int\\:\\frac{e^{2x}\\frac{1}{1+e^{-x}}}{-e^{3x}}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=-\\int\\:\\frac{e^{2x}\\frac{1}{1+e^{-x}}}{e^{3x}}dx"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$",
"secondary": [
"$$\\frac{e^{2x}\\frac{1}{1+e^{-x}}}{e^{3x}}=\\left(e^{2x}\\frac{1}{1+e^{-x}}\\right)e^{-3x}$$"
],
"result": "=-\\int\\:e^{2x}\\frac{1}{1+e^{-x}}e^{-3x}dx",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "Simplify $$e^{2x}\\frac{1}{1+e^{-x}}e^{-3x}:{\\quad}\\frac{e^{-x}}{1+e^{-x}}$$",
"input": "e^{2x}\\frac{1}{1+e^{-x}}e^{-3x}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$e^{2x}e^{-3x}=\\:e^{2x-3x}$$"
],
"result": "=\\frac{1}{1+e^{-x}}e^{2x-3x}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add similar elements: $$2x-3x=-x$$",
"result": "=\\frac{1}{1+e^{-x}}e^{-x}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:e^{-x}}{1+e^{-x}}"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:e^{-x}=e^{-x}$$",
"result": "=\\frac{e^{-x}}{1+e^{-x}}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=-\\int\\:\\frac{e^{-x}}{1+e^{-x}}dx"
},
{
"type": "interim",
"title": "Apply u-substitution",
"input": "\\int\\:\\frac{e^{-x}}{1+e^{-x}}dx",
"steps": [
{
"type": "definition",
"title": "Integral Substitution definition",
"text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$",
"secondary": [
"Substitute: $$u=1+e^{-x}$$"
]
},
{
"type": "interim",
"title": "$$\\frac{du}{dx}=-e^{-x}$$",
"input": "\\left(1+e^{-x}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=1^{^{\\prime}}+\\left(e^{-x}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$1^{\\prime}=0$$",
"input": "1^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/5pHl7jX6nP6oSkaDnJ8dqboRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhNUy0PGUeVurnmhVFT+4jBc="
}
},
{
"type": "interim",
"title": "$$\\left(e^{-x}\\right)^{\\prime}=-e^{-x}$$",
"input": "\\left(e^{-x}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{-x}\\left(-x\\right)^{\\prime}$$",
"input": "\\left(e^{-x}\\right)^{\\prime}",
"result": "=e^{-x}\\left(-x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=-x$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(-x\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(-x\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=-x$$",
"result": "=e^{-x}\\left(-x\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GSIQlKG9310R1ElGl3CVfaRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOB6LdUC4vDg25Wev+HuCzhfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "interim",
"title": "$$\\left(-x\\right)^{\\prime}=-1$$",
"input": "\\left(-x\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=-x^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$x^{\\prime}=1$$",
"result": "=-1"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UM4LDHFZjLzDjY8O/P1Ab8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/g69GyryCsZzNNAEljQyUl0eplVd2gGs/MYc2HGVEvpw+gc+cK9cns8BOox/q2OecQ"
}
},
{
"type": "step",
"result": "=e^{-x}\\left(-1\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-e^{-x}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=0-e^{-x}"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-e^{-x}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:du=-e^{-x}dx$$"
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:dx=\\left(-e^{x}\\right)du$$"
},
{
"type": "step",
"result": "=\\int\\:\\frac{e^{-x}}{u}\\left(-e^{x}\\right)du"
},
{
"type": "interim",
"title": "Simplify $$\\frac{e^{-x}}{u}\\left(-e^{x}\\right):{\\quad}-\\frac{1}{u}$$",
"input": "\\frac{e^{-x}}{u}\\left(-e^{x}\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{e^{-x}}{u}e^{x}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=-\\frac{e^{-x}e^{x}}{u}"
},
{
"type": "interim",
"title": "$$e^{-x}e^{x}=1$$",
"input": "e^{-x}e^{x}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$e^{-x}e^{x}=\\:e^{-x+x}$$"
],
"result": "=e^{-x+x}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add similar elements: $$-x+x=0$$",
"result": "=e^{0}"
},
{
"type": "step",
"primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xTyy5ngoavpgJhDWv/GYNd6GQqufR6tr2vPxOUv7H++P6Ubiv/bIrpol3G9QIK7hIyLGydo+KydlY3bqCHx2y94B8gB1xs8WTDicQ2y4oR0="
}
},
{
"type": "step",
"result": "=-\\frac{1}{u}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{1}{u}du"
}
],
"meta": {
"interimType": "Integral U Substitution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/WOhQiXKi9yVklE5DMKcxeg4hnUrXtr12VFUZl/drzdB42dCZngAt2UhJjMVkTWLsN5vRWJvr0V/8T6RtR0CSAgHP4YqWkh5OOHaSrQ/l8RM0i8HyTLkatLR0zdJkCzvf8DOFkEB41JAUo0zfus0lAEuDOVaQvKofqHoY5jNapscN6G3wC0aNOpX3GMtOsWhImpXFf3SOUx+H18qfp3MLg="
}
},
{
"type": "step",
"result": "=-\\int\\:-\\frac{1}{u}du"
},
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=-\\left(-\\int\\:\\frac{1}{u}du\\right)"
},
{
"type": "step",
"primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right),\\:$$assuming a complex-valued logarithm",
"result": "=-\\left(-\\ln\\left(u\\right)\\right)"
},
{
"type": "step",
"primary": "Substitute back $$u=1+e^{-x}$$",
"result": "=-\\left(-\\ln\\left(1+e^{-x}\\right)\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=\\ln\\left(1+e^{-x}\\right)",
"meta": {
"solvingClass": "Solver"
}
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=\\ln\\left(1+e^{-x}\\right)+C",
"meta": {
"title": {
"extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "step",
"result": "u_{2}=\\ln\\left(1+e^{-x}\\right)"
},
{
"type": "step",
"primary": "$$y_{p}=u_{1}y_{1}+u_{2}y_{2}$$",
"result": "y_{p}=\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}"
},
{
"type": "step",
"primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(x\\right)-3y^{\\prime}\\left(x\\right)+2y=\\frac{1}{1+e^{-x}}{\\quad}$$is:",
"result": "y_{p}=\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}"
}
],
"meta": {
"interimType": "Generic Find That Satisfies Title 2Eq"
}
},
{
"type": "step",
"primary": "The general solution $$y=y_h+y_p$$ is:",
"result": "y=c_{1}e^{2x}+c_{2}e^{x}+\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}"
}
],
"meta": {
"interimType": "ODE Solve Linear 0Eq"
}
},
{
"type": "step",
"result": "y=c_{1}e^{2x}+c_{2}e^{x}+\\left(-1-e^{-x}+\\ln\\left(1+e^{-x}\\right)\\right)e^{2x}+\\ln\\left(1+e^{-x}\\right)e^{x}"
}
],
"meta": {
"solvingClass": "ODE"
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
Solution steps
Solve linear ODE: