y^{''}+4y^'+4y=e^{2x}

{ "query": { "display": "$$y^{^{\\prime\\prime}}+4y^{^{\\prime}}+4y=e^{2x}$$", "symbolab_question": "ODE#y^{\\prime \\prime }+4y^{\\prime }+4y=e^{2x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearNonHomogeneous", "default": "y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\\frac{1}{16}e^{2x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=e^{2x}:{\\quad}y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\\frac{1}{16}e^{2x}$$", "input": "y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=e^{2x}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\\frac{1}{16}e^{2x}$$", "input": "y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=e^{2x}", "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 $$y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=0:{\\quad}y=c_{1}e^{-2x}+c_{2}xe^{-2x}$$", "input": "y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=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}}+4\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime}}+4e^{γx}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γx}\\right)\\right)^{\\prime\\prime}+4\\left(\\left(e^{γx}\\right)\\right)^{\\prime}+4e^{γx}=0:{\\quad}e^{γx}\\left(γ^{2}+4γ+4\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime\\prime}}+4\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime}}+4e^{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}+4\\left(e^{γx}\\right)^{^{\\prime}}+4e^{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}+4e^{γx}γ+4e^{γx}=0" }, { "type": "step", "primary": "Factor $$e^{γx}$$", "result": "e^{γx}\\left(γ^{2}+4γ+4\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γx}\\left(γ^{2}+4γ+4\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γx}\\left(γ^{2}+4γ+4\\right)=0:{\\quad}γ=-2$$ with multiplicity of $$2$$", "input": "e^{γx}\\left(γ^{2}+4γ+4\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γx}\\ne\\:0$$, solving $$e^{γx}\\left(γ^{2}+4γ+4\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}+4γ+4=0$$", "result": "γ^{2}+4γ+4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "γ^{2}+4γ+4=0", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:4}}{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 $${\\quad}a=1,\\:b=4,\\:c=4$$", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:4}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$4^{2}-4\\cdot\\:1\\cdot\\:4=0$$", "input": "4^{2}-4\\cdot\\:1\\cdot\\:4", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{0}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:4=16$$", "result": "=4^{2}-16" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16-16" }, { "type": "step", "primary": "Subtract the numbers: $$16-16=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+7G66pFKfN1rBf+D/KPSDXG8dn4lo8H14604qBXoFzUDnzlbPZjyKgy1eUCFsLd5uA0vbJw0XRhSBCaIjQqkFuHtxdUHQ9/EhXCD2nGKwGhrCM4HC+6g8lrZwhccAADrsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "γ=\\frac{-4}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$\\frac{-4}{2\\cdot\\:1}=-2$$", "input": "\\frac{-4}{2\\cdot\\:1}", "result": "γ=-2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-4}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OPJYQ+MwmOz+kVvGaHPnJ2gelhrK8M2TY4B4ebvtusYJQJZuTAY5js+oqjdT8ksl2wi7sDFXUdWBamMMAmaYmztRXj6DFlQwKiWmso5NKix/8FF0NIjcdEihCunIpTqiJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "γ=-2\\mathrm{\\:with\\:multiplicity\\:of\\:}2" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=-2\\mathrm{\\:with\\:multiplicity\\:of\\:}2" }, { "type": "step", "primary": "For real root $$γ\\:$$with multiplicity $$2,\\:$$the general solution takes the form:$${\\quad}y=c_{1}e^{γ\\:x}+c_{2}xe^{γ\\:x}$$", "result": "y=c_{1}e^{-2x}+c_{2}xe^{-2x}" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Find By Solving Title 2Eq" } }, { "type": "interim", "title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=e^{2x}:{\\quad}y=\\frac{1}{16}e^{2x}$$", "steps": [ { "type": "step", "primary": "For the non-homogeneous part $$g\\left(x\\right)=e^{2x}$$, assume a solution of the form: $$y=a_{0}e^{2x}$$" }, { "type": "step", "result": "\\left(\\left(a_{0}e^{2x}\\right)\\right)^{^{\\prime\\prime}}+4\\left(\\left(a_{0}e^{2x}\\right)\\right)^{^{\\prime}}+4a_{0}e^{2x}=e^{2x}" }, { "type": "interim", "title": "Simplify $$\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime\\prime}+4\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime}+4a_{0}e^{2x}=e^{2x}:{\\quad}16a_{0}e^{2x}=e^{2x}$$", "steps": [ { "type": "interim", "title": "$$\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime\\prime}=4a_{0}e^{2x}$$", "input": "\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime\\prime}", "steps": [ { "type": "interim", "title": "$$\\left(a_{0}e^{2x}\\right)^{\\prime}=a_{0}e^{2x}\\cdot\\:2$$", "input": "\\left(a_{0}e^{2x}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\left(e^{2x}\\right)^{^{\\prime}}" }, { "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": "=a_{0}e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\left(a_{0}e^{2x}\\cdot\\:2\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$\\left(a_{0}e^{2x}\\cdot\\:2\\right)^{\\prime}=4a_{0}e^{2x}$$", "input": "\\left(a_{0}e^{2x}\\cdot\\:2\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\cdot\\:2\\left(e^{2x}\\right)^{^{\\prime}}" }, { "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": "=a_{0}\\cdot\\:2e^{2x}\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "=4a_{0}e^{2x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4a_{0}e^{2x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "4a_{0}e^{2x}+4\\left(\\left(a_{0}e^{2x}\\right)\\right)^{^{\\prime}}+4a_{0}e^{2x}=e^{2x}" }, { "type": "interim", "title": "$$\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime}=a_{0}e^{2x}\\cdot\\:2$$", "input": "\\left(\\left(a_{0}e^{2x}\\right)\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\left(e^{2x}\\right)^{^{\\prime}}" }, { "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": "=a_{0}e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "4a_{0}e^{2x}+4a_{0}e^{2x}\\cdot\\:2+4a_{0}e^{2x}=e^{2x}" }, { "type": "step", "primary": "Simplify", "result": "16a_{0}e^{2x}=e^{2x}" } ], "meta": { "interimType": "ODE Derive And Simplify 0Eq" } }, { "type": "step", "primary": "Find a solution for the coefficient(s) $$a_{0}$$" }, { "type": "interim", "title": "Solve $$16a_{0}e^{2x}=e^{2x}:{\\quad}a_{0}=\\frac{1}{16}$$", "input": "16a_{0}e^{2x}=e^{2x}", "steps": [ { "type": "interim", "title": "Divide both sides by $$16e^{2x}$$", "input": "16a_{0}e^{2x}=e^{2x}", "result": "a_{0}=\\frac{1}{16}", "steps": [ { "type": "step", "primary": "Divide both sides by $$16e^{2x}$$", "result": "\\frac{16a_{0}e^{2x}}{16e^{2x}}=\\frac{e^{2x}}{16e^{2x}}" }, { "type": "step", "primary": "Simplify", "result": "a_{0}=\\frac{1}{16}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "Plug the parameter solutions into $$y=a_{0}e^{2x}$$", "result": "y=\\frac{1}{16}e^{2x}" }, { "type": "step", "primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(x\\right)+4y^{\\prime}\\left(x\\right)+4y=e^{2x}{\\quad}$$is:", "result": "y=\\frac{1}{16}e^{2x}" } ], "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}xe^{-2x}+\\frac{1}{16}e^{2x}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\\frac{1}{16}e^{2x}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\\frac{1}{16}e^{2x}" } } }, "meta": { "showVerify": true } }