{
"query": {
"display": "$$y^{^{\\prime\\prime}}+36y=-60\\sec\\left(6t\\right)$$",
"symbolab_question": "ODE#y^{\\prime \\prime }+36y=-60\\sec(6t)"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "ODE",
"subTopic": "ConstCoeffLinearNonHomogeneous",
"default": "y=c_{1}\\cos(6t)+c_{2}\\sin(6t)-\\frac{5}{3}\\ln(\\cos(6t))\\cos(6t)-10t\\sin(6t)",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$y^{\\prime\\prime}\\left(t\\right)+36y=-60\\sec\\left(6t\\right):{\\quad}y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)+36y=-60\\sec\\left(6t\\right)",
"steps": [
{
"type": "interim",
"title": "Solve linear ODE:$${\\quad}y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)+36y=-60\\sec\\left(6t\\right)",
"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 $$y^{\\prime\\prime}\\left(t\\right)+36y=0:{\\quad}y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)+36y=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 $$e^{γt}$$",
"secondary": [
"Rewrite the equation with $$y=e^{γt}$$"
],
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}+36e^{γt}=0"
},
{
"type": "interim",
"title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}+36e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}+36\\right)=0$$",
"steps": [
{
"type": "step",
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}+36e^{γt}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime}=e^{γt}γ$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\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(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γt}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(e^{γt}γ\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}γ\\right)^{\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}γ\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γ\\left(e^{γt}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\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(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=γe^{γt}γ"
},
{
"type": "interim",
"title": "Simplify $$γe^{γt}γ:{\\quad}γ^{2}e^{γt}$$",
"input": "γe^{γt}γ",
"result": "=γ^{2}e^{γt}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$γγ=\\:γ^{1+1}$$"
],
"result": "=e^{γt}γ^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=e^{γt}γ^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7riofKsRQ/FkpZv0BW2pxMd6GQqufR6tr2vPxOUv7H+/BWItNlNCsjK5QfFqGTa8umx4rCXhbsN+br+uOYP22UU3kCh3oevUunZ7/b0qFKBStCRMtul5SOs/SBwPTbaWuo4bl40YraHWFXpFVaYGPXg=="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=γ^{2}e^{γt}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γt}+36e^{γt}=0"
},
{
"type": "step",
"primary": "Factor $$e^{γt}$$",
"result": "e^{γt}\\left(γ^{2}+36\\right)=0"
}
],
"meta": {
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "step",
"result": "e^{γt}\\left(γ^{2}+36\\right)=0"
},
{
"type": "interim",
"title": "Solve $$e^{γt}\\left(γ^{2}+36\\right)=0:{\\quad}γ=6i,\\:γ=-6i$$",
"input": "e^{γt}\\left(γ^{2}+36\\right)=0",
"steps": [
{
"type": "step",
"primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}+36\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}+36=0$$",
"result": "γ^{2}+36=0"
},
{
"type": "interim",
"title": "Move $$36\\:$$to the right side",
"input": "γ^{2}+36=0",
"result": "γ^{2}=-36",
"steps": [
{
"type": "step",
"primary": "Subtract $$36$$ from both sides",
"result": "γ^{2}+36-36=0-36"
},
{
"type": "step",
"primary": "Simplify",
"result": "γ^{2}=-36"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
},
{
"type": "step",
"primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$"
},
{
"type": "step",
"result": "γ=\\sqrt{-36},\\:γ=-\\sqrt{-36}"
},
{
"type": "interim",
"title": "$$\\sqrt{-36}=6i$$",
"input": "\\sqrt{-36}",
"steps": [
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{a}\\sqrt{-1}$$",
"secondary": [
"$$\\sqrt{-36}=\\sqrt{36}\\sqrt{-1}$$"
],
"result": "=\\sqrt{36}\\sqrt{-1}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$",
"result": "=\\sqrt{36}i",
"meta": {
"practiceLink": "/practice/complex-numbers-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\sqrt{36}=6$$",
"input": "\\sqrt{36}",
"steps": [
{
"type": "step",
"primary": "Factor the number: $$36=6^{2}$$",
"result": "=\\sqrt{6^{2}}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$",
"secondary": [
"$$\\sqrt{6^{2}}=6$$"
],
"result": "=6"
}
],
"meta": {
"solvingClass": "Solver2",
"interimType": "Solver2",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ba5g+UZ5roMy68ii/TX6lSAn9lkDfZkicUGkO3EF+IowGPWQQxj+Buy70vvMPVbRzGQBGgzjhY9wy62Ypaxu7DP9zNOAin/+9QrppOujlp4="
}
},
{
"type": "step",
"result": "=6i"
}
],
"meta": {
"solvingClass": "Solver2",
"interimType": "Solver2",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oGU6sXD51uYqrDyiC7DJKN6GQqufR6tr2vPxOUv7H+9bQPAhKkRHwQtN722Vq170hj6ylSwHjJkmQPKH34+7f/Q9J8iednviAQ8XqKUxpSA="
}
},
{
"type": "interim",
"title": "$$-\\sqrt{-36}=-6i$$",
"input": "-\\sqrt{-36}",
"steps": [
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{a}\\sqrt{-1}$$",
"secondary": [
"$$-\\sqrt{-36}=-\\sqrt{36}\\sqrt{-1}$$"
],
"result": "=-\\sqrt{36}\\sqrt{-1}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$",
"result": "=-\\sqrt{36}i",
"meta": {
"practiceLink": "/practice/complex-numbers-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\sqrt{36}=6$$",
"input": "\\sqrt{36}",
"steps": [
{
"type": "step",
"primary": "Factor the number: $$36=6^{2}$$",
"result": "=\\sqrt{6^{2}}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$",
"secondary": [
"$$\\sqrt{6^{2}}=6$$"
],
"result": "=6"
}
],
"meta": {
"solvingClass": "Solver2",
"interimType": "Solver2",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ba5g+UZ5roMy68ii/TX6lSAn9lkDfZkicUGkO3EF+IowGPWQQxj+Buy70vvMPVbRzGQBGgzjhY9wy62Ypaxu7DP9zNOAin/+9QrppOujlp4="
}
},
{
"type": "step",
"result": "=-6i"
}
],
"meta": {
"solvingClass": "Solver2",
"interimType": "Solver2",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A/rlP7B78aIdO/PL1Tz02VXTSum/z5kLpMzXS1UJIeyj0KG0lvFYbYVov0nziiPsCXP3HRkwvQo8XuRFa/NQF7WRjDkGr5D0b3lZyvYbjrUkt3WiGR7ZaCaXvz77bMjS"
}
},
{
"type": "step",
"result": "γ=6i,\\:γ=-6i"
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "γ=6i,\\:γ=-6i"
},
{
"type": "step",
"primary": "For two complex roots $$γ_{1}\\ne\\:γ_{2}$$, where $$γ_{1}=\\alpha+i\\:\\beta,\\:γ_{2}=\\alpha-i\\:\\beta\\:$$<br/>the general solution takes the form:$${\\quad}y=e^{\\alpha\\:t}\\left(c_{1}\\cos\\left(\\beta\\:t\\right)+c_{2}\\sin\\left(\\beta\\:t\\right)\\right)$$",
"result": "e^{0}\\left(c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)"
}
],
"meta": {
"solvingClass": "ODE",
"interimType": "Generic Find By Solving Title 2Eq"
}
},
{
"type": "interim",
"title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(t\\right)+36y=-60\\sec\\left(6t\\right):{\\quad}y_{p}=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)$$",
"steps": [
{
"type": "step",
"primary": "For the non-homogeneous part $$g\\left(t\\right)=-60\\sec\\left(6t\\right)$$, 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/>$$\\begin{pmatrix}\\left(u_{1}\\right)^{^{\\prime}}y_{1}+\\left(u_{2}\\right)^{^{\\prime}}y_{2}=0\\\\\\left(u_{1}\\right)^{^{\\prime}}\\left(y_{1}\\right)^{^{\\prime}}+\\left(u_{2}\\right)^{^{\\prime}}\\left(y_{2}\\right)^{^{\\prime}}=g\\left(t\\right)\\end{pmatrix}$$<br/>Which implies",
"secondary": [
"$$u_{1}=\\int\\:-\\frac{y_{2}g\\left(t\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dt$$",
"$$u_{2}=\\int\\:\\frac{y_{1}g\\left(t\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dt$$",
"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": [
"$$y_{1}=\\cos\\left(6t\\right)$$",
"$$y_{2}=\\sin\\left(6t\\right)$$"
]
},
{
"type": "interim",
"title": "$$\\left(y_{1}\\right)^{^{\\prime}}:{\\quad}-\\sin\\left(6t\\right)\\cdot\\:6$$",
"input": "\\left(\\cos\\left(6t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}-\\sin\\left(6t\\right)\\left(6t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(6t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(6t\\right)\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=6t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(6t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(6t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=6t$$",
"result": "=-\\sin\\left(6t\\right)\\left(6t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyIu/GYkmDFNPqT12rKBwQ9JJQz2lqSQogu9PoWz88zfnZQ9Z3Siv7iLdHqZDfvh55mmSkC3Mw9EDaBd1JKNjy8RLUkD296vqd81SRk8bq7AlJcj6EQVgLle9yFp8cy8qck3kCh3oevUunZ7/b0qFKBSgKzM+BlIedhng7m5DQQLWtIFBygoY9STUrg+h4aNkjw=="
}
},
{
"type": "interim",
"title": "$$\\left(6t\\right)^{\\prime}=6$$",
"input": "\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=6t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=6\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=6",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zgsQLHKcE4bPMk/tfhJr3MPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gV29r656qFR0DUTLPqu9PtMnS+huczlXBlThLcwNi14o7wEWcvxYk7LAulItRhl3H"
}
},
{
"type": "step",
"result": "=-\\sin\\left(6t\\right)\\cdot\\:6"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(y_{2}\\right)^{^{\\prime}}:{\\quad}\\cos\\left(6t\\right)\\cdot\\:6$$",
"input": "\\left(\\sin\\left(6t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}\\cos\\left(6t\\right)\\left(6t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(6t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(6t\\right)\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=6t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(6t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(6t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=6t$$",
"result": "=\\cos\\left(6t\\right)\\left(6t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przYu/GYkmDFNPqT12rKBwQ9JJQz2lqSQogu9PoWz88zfnZQ9Z3Siv7iLdHqZDfvh55mmSkC3Mw9EDaBd1JKNjy8STo/r9qs40dtZcwt9pfTzzV9kLmPuggiAJDvGg3AtI9KqQhfqSx4Ip0S2H5wgufDIxf2pgqTS+Ia9coexBJUV6iL5k+VDooFqbSQg5EE31pQ=="
}
},
{
"type": "interim",
"title": "$$\\left(6t\\right)^{\\prime}=6$$",
"input": "\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=6t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=6\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=6",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zgsQLHKcE4bPMk/tfhJr3MPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gV29r656qFR0DUTLPqu9PtMnS+huczlXBlThLcwNi14o7wEWcvxYk7LAulItRhl3H"
}
},
{
"type": "step",
"result": "=\\cos\\left(6t\\right)\\cdot\\:6"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"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)=\\cos\\left(6t\\right)\\cos\\left(6t\\right)\\cdot\\:6-\\left(-\\sin\\left(6t\\right)\\cdot\\:6\\right)\\sin\\left(6t\\right)"
},
{
"type": "step",
"result": "W\\left(y_{1},\\:y_{2}\\right)=6"
},
{
"type": "step",
"primary": "$$u_{1}=\\int\\:-\\frac{y_{2}g\\left(t\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dt$$",
"result": "u_{1}=\\int\\:-\\frac{\\sin\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt"
},
{
"type": "interim",
"title": "$$\\int\\:-\\frac{\\sin\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)+C$$",
"input": "\\int\\:-\\frac{\\sin\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt",
"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": "=-\\frac{1}{6}\\cdot\\:\\int\\:\\sin\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)dt"
},
{
"type": "interim",
"title": "Rewrite using trig identities",
"input": "\\int\\:\\sin\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)dt",
"result": "=-\\frac{1}{6}\\cdot\\:\\int\\:-\\frac{60\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}dt",
"steps": [
{
"type": "step",
"primary": "Use the basic trigonometric identity: $$\\sec\\left(x\\right)=\\frac{1}{\\cos\\left(x\\right)}$$",
"result": "=\\int\\:\\sin\\left(6t\\right)\\left(-60\\cdot\\:\\frac{1}{\\cos\\left(6t\\right)}\\right)dt"
},
{
"type": "interim",
"title": "Simplify $$\\sin\\left(6t\\right)\\left(-60\\cdot\\:\\frac{1}{\\cos\\left(6t\\right)}\\right):{\\quad}-\\frac{60\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}$$",
"input": "\\sin\\left(6t\\right)\\left(-60\\cdot\\:\\frac{1}{\\cos\\left(6t\\right)}\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\sin\\left(6t\\right)\\cdot\\:60\\cdot\\:\\frac{1}{\\cos\\left(6t\\right)}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=-\\frac{1\\cdot\\:\\sin\\left(6t\\right)\\cdot\\:60}{\\cos\\left(6t\\right)}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:60=60$$",
"result": "=-\\frac{60\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{60\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}dt"
}
],
"meta": {
"interimType": "Trig Rewrite Using Trig identities 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/Ho1f5syByYLX5P2O73/2j4JMNHLEeqSQIwGXmmBIh+PosKaRMq/pBPumoBrLj89/lZJuHdgx059G6Z6iwp6k/6DZLE6vHKrQJpnGEY59NTD+ZwthOp2oPXI6jTaq4Gz01OtTMuHZvTePg2+GYbCdMIVCLSxSy8rv0jRguSUmNCZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA=="
}
},
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=-\\frac{1}{6}\\left(-60\\cdot\\:\\int\\:\\frac{\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}dt\\right)"
},
{
"type": "interim",
"title": "Apply u-substitution",
"input": "\\int\\:\\frac{\\sin\\left(6t\\right)}{\\cos\\left(6t\\right)}dt",
"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: $$u=\\cos\\left(6t\\right)$$"
]
},
{
"type": "interim",
"title": "$$\\frac{du}{dt}=-\\sin\\left(6t\\right)6$$",
"input": "\\left(\\cos\\left(6t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}-\\sin\\left(6t\\right)\\left(6t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(6t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(6t\\right)\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=6t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(6t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(6t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=6t$$",
"result": "=-\\sin\\left(6t\\right)\\left(6t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyIu/GYkmDFNPqT12rKBwQ9JJQz2lqSQogu9PoWz88zfnZQ9Z3Siv7iLdHqZDfvh55mmSkC3Mw9EDaBd1JKNjy8RLUkD296vqd81SRk8bq7AlJcj6EQVgLle9yFp8cy8qck3kCh3oevUunZ7/b0qFKBSgKzM+BlIedhng7m5DQQLWtIFBygoY9STUrg+h4aNkjw=="
}
},
{
"type": "interim",
"title": "$$\\left(6t\\right)^{\\prime}=6$$",
"input": "\\left(6t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=6t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=6\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=6",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zgsQLHKcE4bPMk/tfhJr3MPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gV29r656qFR0DUTLPqu9PtMnS+huczlXBlThLcwNi14o7wEWcvxYk7LAulItRhl3H"
}
},
{
"type": "step",
"result": "=-\\sin\\left(6t\\right)\\cdot\\:6"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:du=-\\sin\\left(6t\\right)6dt$$"
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:dt=\\left(-\\frac{1}{6\\sin\\left(6t\\right)}\\right)du$$"
},
{
"type": "step",
"result": "=\\int\\:\\frac{\\sin\\left(6t\\right)}{u}\\left(-\\frac{1}{6\\sin\\left(6t\\right)}\\right)du"
},
{
"type": "interim",
"title": "Simplify $$\\frac{\\sin\\left(6t\\right)}{u}\\left(-\\frac{1}{6\\sin\\left(6t\\right)}\\right):{\\quad}-\\frac{1}{6u}$$",
"input": "\\frac{\\sin\\left(6t\\right)}{u}\\left(-\\frac{1}{6\\sin\\left(6t\\right)}\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{\\sin\\left(6t\\right)}{u}\\cdot\\:\\frac{1}{6\\sin\\left(6t\\right)}"
},
{
"type": "step",
"primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$",
"result": "=-\\frac{\\sin\\left(6t\\right)\\cdot\\:1}{u\\cdot\\:6\\sin\\left(6t\\right)}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$\\sin\\left(6t\\right)$$",
"result": "=-\\frac{1}{u\\cdot\\:6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\int\\:-\\frac{1}{6u}du"
}
],
"meta": {
"interimType": "Integral U Substitution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70pAj3nshv2/cgcf2w7gObzU0lGg8OCCWpWoa+bX9fNNpN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqiXhnWZkvWO3GkAy05vuDu098kl0JLOd5E4HERWWY8WYR6pfF1z6umzUJTJvt+ojYZJI7syIHiVINdMmyoKtK9trcRCHe5WXEh3UwLAL5k2RY="
}
},
{
"type": "step",
"result": "=-\\frac{1}{6}\\left(-60\\cdot\\:\\int\\:-\\frac{1}{6u}du\\right)"
},
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=-\\frac{1}{6}\\left(-60\\left(-\\frac{1}{6}\\cdot\\:\\int\\:\\frac{1}{u}du\\right)\\right)"
},
{
"type": "step",
"primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right),\\:$$assuming a complex-valued logarithm",
"result": "=-\\frac{1}{6}\\left(-60\\left(-\\frac{1}{6}\\ln\\left(u\\right)\\right)\\right)"
},
{
"type": "step",
"primary": "Substitute back $$u=\\cos\\left(6t\\right)$$",
"result": "=-\\frac{1}{6}\\left(-60\\left(-\\frac{1}{6}\\ln\\left(\\cos\\left(6t\\right)\\right)\\right)\\right)"
},
{
"type": "interim",
"title": "Simplify $$-\\frac{1}{6}\\left(-60\\left(-\\frac{1}{6}\\ln\\left(\\cos\\left(6t\\right)\\right)\\right)\\right):{\\quad}-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)$$",
"input": "-\\frac{1}{6}\\left(-60\\left(-\\frac{1}{6}\\ln\\left(\\cos\\left(6t\\right)\\right)\\right)\\right)",
"result": "=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$",
"result": "=-\\frac{1}{6}\\cdot\\:60\\cdot\\:\\frac{1}{6}\\ln\\left(\\cos\\left(6t\\right)\\right)"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$",
"result": "=-\\frac{1\\cdot\\:1\\cdot\\:60}{6\\cdot\\:6}\\ln\\left(\\cos\\left(6t\\right)\\right)"
},
{
"type": "interim",
"title": "$$\\frac{1\\cdot\\:1\\cdot\\:60}{6\\cdot\\:6}=\\frac{5}{3}$$",
"input": "\\frac{1\\cdot\\:1\\cdot\\:60}{6\\cdot\\:6}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:1\\cdot\\:60=60$$",
"result": "=\\frac{60}{6\\cdot\\:6}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$6\\cdot\\:6=36$$",
"result": "=\\frac{60}{36}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$12$$",
"result": "=\\frac{5}{3}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCowE2URjZARlATUv6zVtxJNE7r4fTGiuAq5JqOECsqeiCUCWbkwGOY7PqKo3U/JLJep+4H7LyhX9URx2zSV8Ceo/y9DKGIPglJ+qMi9xDu2K0Y8qUu7oNXob/DVNHRtUnpJqWSvpKvaIS+CLplmxaYOg9OBfnneqJX8MHqhtbw44V2XjVNEekh8JtG3ZXDlTQQ=="
}
},
{
"type": "step",
"result": "=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dEuqhqh+PfYcuFG0/ZQwNDqZekvJjcQK6OkLUxJHCxkpAo1lvc7yWNp4VIEbivA89s43/aG4RqIhR8wn5cy6THm0yklMEFxxRaRLK8x7VKkyB9dQ6271ePBbbMIuwSyP0v0ULx2e3D9Bw5jDhjQWlYEFMST8lDZxn1Yq5HMKVTvaBf/y/iatBPxn+bAoT0bW1++P4VvM3vd7qongLhcLBT8GHcxh/HLhbnEHrfyuvOQtS4B8vIL2NCshVI1KoKTIJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\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}=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)"
},
{
"type": "step",
"primary": "$$u_{2}=\\int\\:\\frac{y_{1}g\\left(t\\right)}{W\\left(y_{1},\\:y_{2}\\right)}dt$$",
"result": "u_{2}=\\int\\:\\frac{\\cos\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt"
},
{
"type": "interim",
"title": "$$\\int\\:\\frac{\\cos\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt=-10t+C$$",
"input": "\\int\\:\\frac{\\cos\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}dt",
"steps": [
{
"type": "interim",
"title": "$$\\frac{\\cos\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}=-10$$",
"input": "\\frac{\\cos\\left(6t\\right)\\left(-60\\sec\\left(6t\\right)\\right)}{6}",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=\\frac{-\\cos\\left(6t\\right)\\cdot\\:60\\sec\\left(6t\\right)}{6}"
},
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$",
"result": "=-\\frac{\\cos\\left(6t\\right)\\cdot\\:60\\sec\\left(6t\\right)}{6}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{60}{6}=10$$",
"result": "=-10\\cos\\left(6t\\right)\\sec\\left(6t\\right)"
},
{
"type": "interim",
"title": "$$\\sec\\left(6t\\right)\\cos\\left(6t\\right)=1$$",
"input": "\\sec\\left(6t\\right)\\cos\\left(6t\\right)",
"steps": [
{
"type": "interim",
"title": "Express with sin, cos",
"input": "\\sec\\left(6t\\right)\\cos\\left(6t\\right)",
"steps": [
{
"type": "step",
"primary": "Use the basic trigonometric identity: $$\\sec\\left(6t\\right)=\\frac{1}{\\cos\\left(6t\\right)}$$",
"result": "=\\frac{1}{\\cos\\left(6t\\right)}\\cos\\left(6t\\right)"
},
{
"type": "interim",
"title": "$$\\frac{1}{\\cos\\left(6t\\right)}\\cos\\left(6t\\right)=1$$",
"input": "\\frac{1}{\\cos\\left(6t\\right)}\\cos\\left(6t\\right)",
"result": "=1",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cos\\left(6t\\right)}{\\cos\\left(6t\\right)}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$\\cos\\left(6t\\right)$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s743+Hzh+1NcELvtHIMKV2UMgqTRDXkrDedgMXPjnHlLotOtZYwUjyXhDTsNnn6ElrPGrllHeFjaKLCz4xpIb40UeCBKuYKgaNJ253gLI69U75pEyVJwsO/EEQdwGeaLTMNZInpLv7ge9+QPnPK/GU8Q=="
}
}
],
"meta": {
"interimType": "Trig Express Sin Cos 0Eq"
}
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=-10\\cdot\\:1"
},
{
"type": "step",
"primary": "Multiply the numbers: $$10\\cdot\\:1=10$$",
"result": "=-10"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UBFdZJfM6LtMHjZGPeyr3c101Rb0EKrI1B4fDzRx0T+axcbbZAOSNlm7F86qWQtXzMFYmi1F5Hg/ibpEToVnY8LDiesSbIvTYqz2p2k8mWaFAx1vVBfYA0EqU/MNM02oWyczvjuiETmqRwoz9fEzFEgTNeDaE3f6yTz7Ce+SWdc="
}
},
{
"type": "step",
"result": "=\\int\\:-10dt"
},
{
"type": "step",
"primary": "Integral of a constant: $$\\int{a}dx=ax$$",
"result": "=\\left(-10\\right)t"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-10t",
"meta": {
"solvingClass": "Solver"
}
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=-10t+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}=-10t"
},
{
"type": "step",
"primary": "$$y_{p}=u_{1}y_{1}+u_{2}y_{2}$$",
"result": "y_{p}=\\left(-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\right)\\cos\\left(6t\\right)+\\left(-10t\\right)\\sin\\left(6t\\right)"
},
{
"type": "interim",
"title": "Simplify",
"input": "\\left(-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\right)\\cos\\left(6t\\right)+\\left(-10t\\right)\\sin\\left(6t\\right)",
"result": "y_{p}=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Title 0Eq"
}
},
{
"type": "step",
"primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(t\\right)+36y=-60\\sec\\left(6t\\right){\\quad}$$is:",
"result": "y_{p}=-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)"
}
],
"meta": {
"interimType": "Generic Find That Satisfies Title 2Eq"
}
},
{
"type": "step",
"primary": "The general solution $$y=y_h+y_p$$ is:",
"result": "y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)"
}
],
"meta": {
"interimType": "ODE Solve Linear 0Eq"
}
},
{
"type": "step",
"result": "y=c_{1}\\cos\\left(6t\\right)+c_{2}\\sin\\left(6t\\right)-\\frac{5}{3}\\ln\\left(\\cos\\left(6t\\right)\\right)\\cos\\left(6t\\right)-10t\\sin\\left(6t\\right)"
}
],
"meta": {
"solvingClass": "ODE"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "t",
"plotRequest": "#>#ODE#>#y=c_{1}\\cos(6t)+c_{2}\\sin(6t)-\\frac{5}{3}\\ln(\\cos(6t))\\cos(6t)-10t\\sin(6t)"
}
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
Solution steps
Solve linear ODE: