y^{''}+4y=e^{(-t)}

{ "query": { "display": "$$y^{^{\\prime\\prime}}+4y=e^{\\left(-t\\right)}$$", "symbolab_question": "ODE#y^{\\prime \\prime }+4y=e^{(-t)}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearNonHomogeneous", "default": "y=c_{1}\\cos(2t)+c_{2}\\sin(2t)+\\frac{1}{5}e^{-t}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(t\\right)+4y=e^{\\left(-t\\right)}:{\\quad}y=c_{1}\\cos\\left(2t\\right)+c_{2}\\sin\\left(2t\\right)+\\frac{1}{5}e^{-t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)+4y=e^{\\left(-t\\right)}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=c_{1}\\cos\\left(2t\\right)+c_{2}\\sin\\left(2t\\right)+\\frac{1}{5}e^{-t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)+4y=e^{\\left(-t\\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 $$y^{\\prime\\prime}\\left(t\\right)+4y=0:{\\quad}y=c_{1}\\cos\\left(2t\\right)+c_{2}\\sin\\left(2t\\right)$$", "input": "y^{\\prime\\prime}\\left(t\\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 $$e^{γt}$$", "secondary": [ "Rewrite the equation with $$y=e^{γt}$$" ], "result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}+4e^{γt}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}+4e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}+4\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}+4e^{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}+4e^{γt}=0" }, { "type": "step", "primary": "Factor $$e^{γt}$$", "result": "e^{γt}\\left(γ^{2}+4\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γt}\\left(γ^{2}+4\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γt}\\left(γ^{2}+4\\right)=0:{\\quad}γ=2i,\\:γ=-2i$$", "input": "e^{γt}\\left(γ^{2}+4\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}+4\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}+4=0$$", "result": "γ^{2}+4=0" }, { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "γ^{2}+4=0", "result": "γ^{2}=-4", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "γ^{2}+4-4=0-4" }, { "type": "step", "primary": "Simplify", "result": "γ^{2}=-4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ma8ufBc9jY6seNp7JCqu2z/R6KHj8NkmShU5qv4YP6m1qWD1tj0yCrqFZUoN1F0S2aDA1+ZGUCYqL+OrRf9TbbKcMR1HkepIx+jM2lhXxXPpdehqXrjoDVji0jK1zIrfU9ik5WINGrsWotLzS7/BjMilG2DUqMrP9Xo55x7KtrDbW1CT6iTiD10pfBbS+/CQpvNnq+4D/2wGwpTtqV+k2v1mKRo6wqmQcIJrGZMsEPin6WU7ybUlSwezFhLKvkKNVvuqUoYkKmBK6fSR2cNlgkfqfT3HSxldmNt6rUcfvySCyrAO9zMQDZyjEiWM7qI5HNLamESyiXkEVNU4Be3yXlX3rP7J381OCLdf7C7WWRDvbBmbuQNTF0TphKZ8RuvaLxm++OQazgp4n/OzBIRt/VHyA5GEFWW3LlP4nBK806VetYVYdLe45VxLoQDaQoEh" } }, { "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{-4},\\:γ=-\\sqrt{-4}" }, { "type": "interim", "title": "$$\\sqrt{-4}=2i$$", "input": "\\sqrt{-4}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{a}\\sqrt{-1}$$", "secondary": [ "$$\\sqrt{-4}=\\sqrt{4}\\sqrt{-1}$$" ], "result": "=\\sqrt{4}\\sqrt{-1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=\\sqrt{4}i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=2i" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wxohaeJaCzdlhss8MNmRxCAn9lkDfZkicUGkO3EF+IqThYrY0DSIwEtMrt7VRP5Pu0JHl0iPFDkBKD5k8VVKx9ScfzaA6+eQOV183yLHwkI=" } }, { "type": "interim", "title": "$$-\\sqrt{-4}=-2i$$", "input": "-\\sqrt{-4}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{a}\\sqrt{-1}$$", "secondary": [ "$$-\\sqrt{-4}=-\\sqrt{4}\\sqrt{-1}$$" ], "result": "=-\\sqrt{4}\\sqrt{-1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=-\\sqrt{4}i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a^2}=a,\\:\\quad\\:a\\ge0$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c=" } }, { "type": "step", "result": "=-2i" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YG82aXwfXJQaywYBnj1JUd6GQqufR6tr2vPxOUv7H+/Rgy9XNxUSgx9oUFMaW2iN1tujRJib+F5jD+3+fXhwONEgxql6+ZjKJW5GYNxPQK8=" } }, { "type": "step", "result": "γ=2i,\\:γ=-2i" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=2i,\\:γ=-2i" }, { "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(2t\\right)+c_{2}\\sin\\left(2t\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "y=c_{1}\\cos\\left(2t\\right)+c_{2}\\sin\\left(2t\\right)" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Find By Solving Title 2Eq" } }, { "type": "interim", "title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(t\\right)+4y=e^{-t}:{\\quad}y=\\frac{1}{5}e^{-t}$$", "steps": [ { "type": "step", "primary": "For the non-homogeneous part $$g\\left(x\\right)=e^{-t}$$, assume a solution of the form: $$y=a_{0}e^{-t}$$" }, { "type": "step", "result": "\\left(\\left(a_{0}e^{-t}\\right)\\right)^{^{\\prime\\prime}}+4a_{0}e^{-t}=e^{-t}" }, { "type": "interim", "title": "Simplify $$\\left(\\left(a_{0}e^{-t}\\right)\\right)^{\\prime\\prime}+4a_{0}e^{-t}=e^{-t}:{\\quad}5a_{0}e^{-t}=e^{-t}$$", "steps": [ { "type": "interim", "title": "$$\\left(\\left(a_{0}e^{-t}\\right)\\right)^{\\prime\\prime}=a_{0}e^{-t}$$", "input": "\\left(\\left(a_{0}e^{-t}\\right)\\right)^{\\prime\\prime}", "steps": [ { "type": "interim", "title": "$$\\left(a_{0}e^{-t}\\right)^{\\prime}=-a_{0}e^{-t}$$", "input": "\\left(a_{0}e^{-t}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\left(e^{-t}\\right)^{^{\\prime}}" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{-t}\\left(-t\\right)^{\\prime}$$", "input": "\\left(e^{-t}\\right)^{\\prime}", "result": "=e^{-t}\\left(-t\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=-t$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(-t\\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(-t\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=-t$$", "result": "=e^{-t}\\left(-t\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HhqChl4KvVnpfwPWF6ANOaRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQMQLnLmfs8lYfVx7B5/SSGqfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(-t\\right)^{\\prime}=-1$$", "input": "\\left(-t\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-t^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$t^{\\prime}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7geOGFi+AsYgzBmsq21ADaMPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/g69GyryCsZzNNAEljQyUl0aXIoMMRVjN9bqtx4oSL5oCgc+cK9cns8BOox/q2OecQ" } }, { "type": "step", "result": "=a_{0}e^{-t}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-a_{0}e^{-t}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\left(-a_{0}e^{-t}\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$\\left(-a_{0}e^{-t}\\right)^{\\prime}=a_{0}e^{-t}$$", "input": "\\left(-a_{0}e^{-t}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-a_{0}\\left(e^{-t}\\right)^{^{\\prime}}" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{-t}\\left(-t\\right)^{\\prime}$$", "input": "\\left(e^{-t}\\right)^{\\prime}", "result": "=e^{-t}\\left(-t\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=-t$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(-t\\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(-t\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=-t$$", "result": "=e^{-t}\\left(-t\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HhqChl4KvVnpfwPWF6ANOaRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQMQLnLmfs8lYfVx7B5/SSGqfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(-t\\right)^{\\prime}=-1$$", "input": "\\left(-t\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-t^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$t^{\\prime}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7geOGFi+AsYgzBmsq21ADaMPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/g69GyryCsZzNNAEljQyUl0aXIoMMRVjN9bqtx4oSL5oCgc+cK9cns8BOox/q2OecQ" } }, { "type": "step", "result": "=-a_{0}e^{-t}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=a_{0}e^{-t}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=a_{0}e^{-t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "a_{0}e^{-t}+4a_{0}e^{-t}=e^{-t}" }, { "type": "step", "primary": "Simplify", "result": "5a_{0}e^{-t}=e^{-t}" } ], "meta": { "interimType": "ODE Derive And Simplify 0Eq" } }, { "type": "step", "primary": "Find a solution for the coefficient(s) $$a_{0}$$" }, { "type": "interim", "title": "Solve $$5a_{0}e^{-t}=e^{-t}:{\\quad}a_{0}=\\frac{1}{5}$$", "input": "5a_{0}e^{-t}=e^{-t}", "steps": [ { "type": "interim", "title": "Divide both sides by $$5e^{-t}$$", "input": "5a_{0}e^{-t}=e^{-t}", "result": "a_{0}=\\frac{1}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5e^{-t}$$", "result": "\\frac{5a_{0}e^{-t}}{5e^{-t}}=\\frac{e^{-t}}{5e^{-t}}" }, { "type": "step", "primary": "Simplify", "result": "a_{0}=\\frac{1}{5}" } ], "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 $$y=a_{0}e^{-t}$$", "result": "y=\\frac{1}{5}e^{-t}" }, { "type": "step", "primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(t\\right)+4y=e^{-t}{\\quad}$$is:", "result": "y=\\frac{1}{5}e^{-t}" } ], "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(2t\\right)+c_{2}\\sin\\left(2t\\right)+\\frac{1}{5}e^{-t}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=c_{1}\\cos\\left(2t\\right)+c_{2}\\sin\\left(2t\\right)+\\frac{1}{5}e^{-t}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "#>#ODE#>#y=c_{1}\\cos(2t)+c_{2}\\sin(2t)+\\frac{1}{5}e^{-t}" } } }, "meta": { "showVerify": true } }