y^{''}+4y^'=10e^t

{ "query": { "display": "$$y^{^{\\prime\\prime}}+4y^{^{\\prime}}=10e^{t}$$", "symbolab_question": "ODE#y^{\\prime \\prime }+4y^{\\prime }=10e^{t}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearNonHomogeneous", "default": "y=c_{1}+c_{2}e^{-4t}+2e^{t}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=10e^{t}:{\\quad}y=c_{1}+c_{2}e^{-4t}+2e^{t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=10e^{t}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=c_{1}+c_{2}e^{-4t}+2e^{t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=10e^{t}", "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 $$y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=0:{\\quad}y=c_{1}+c_{2}e^{-4t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=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}}+4\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}+4\\left(\\left(e^{γt}\\right)\\right)^{\\prime}=0:{\\quad}e^{γt}\\left(γ^{2}+4γ\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}+4\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}=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}+4\\left(e^{γt}\\right)^{^{\\prime}}=0" }, { "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": "γ^{2}e^{γt}+4e^{γt}γ=0" }, { "type": "step", "primary": "Factor $$e^{γt}$$", "result": "e^{γt}\\left(γ^{2}+4γ\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γt}\\left(γ^{2}+4γ\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γt}\\left(γ^{2}+4γ\\right)=0:{\\quad}γ=0,\\:γ=-4$$", "input": "e^{γt}\\left(γ^{2}+4γ\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}+4γ\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}+4γ=0$$", "result": "γ^{2}+4γ=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "γ^{2}+4γ=0", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:0}}{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 $${\\quad}a=1,\\:b=4,\\:c=0$$", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:0}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:0}=4$$", "input": "\\sqrt{4^{2}-4\\cdot\\:1\\cdot\\:0}", "result": "{γ}_{1,\\:2}=\\frac{-4\\pm\\:4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{4^{2}-0}" }, { "type": "step", "primary": "$$4^{2}-0=4^{2}$$", "result": "=\\sqrt{4^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=4", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74jz8pBgj569lwUyDg0Z0NuLa5CW27RHLIqeo2kEVRqx8kR7hsO/rTOTBE0w4+r1RlFYe8lM/WYPz8s80IajfKGRLd2VwIqlBNByF6663syTlbd5UL5ZztzwKHGYuXpz+yTXCuSV9SCAfXhEhDUvxkrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{γ}_{1}=\\frac{-4+4}{2\\cdot\\:1},\\:{γ}_{2}=\\frac{-4-4}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$γ=\\frac{-4+4}{2\\cdot\\:1}:{\\quad}0$$", "input": "\\frac{-4+4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-4+4=0$$", "result": "=\\frac{0}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TOTOffKBmJMmI/jfMcfP0yJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5uA0vbJw0XRhSBCaIjQqkFmTSU813ZtyJEXUt6tG8aIa85OEcwDjsSXoI6D5lNNGoialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$γ=\\frac{-4-4}{2\\cdot\\:1}:{\\quad}-4$$", "input": "\\frac{-4-4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-4-4=-8$$", "result": "=\\frac{-8}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-8}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{8}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{2}=4$$", "result": "=-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pzqSJHmg+TzoMyix2qkJxyJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5atajFbLiWAEERGJa9D8RpBerdkx4fh/64hGjtYuWxjaU8i8TDFI4RyTJvi4fzpNTbGUhD9+gDQg079F600Xu5Q==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "γ=0,\\:γ=-4" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=0,\\:γ=-4" }, { "type": "step", "primary": "For two real roots $$γ_{1}\\ne\\:γ_{2}$$, the general solution takes the form:$${\\quad}y=c_{1}e^{γ_{1}\\:t}+c_{2}e^{γ_{2}\\:t}$$", "result": "c_{1}e^{0}+c_{2}e^{-4t}" }, { "type": "step", "primary": "Simplify", "result": "y=c_{1}+c_{2}e^{-4t}" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Find By Solving Title 2Eq" } }, { "type": "interim", "title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=10e^{t}:{\\quad}y=2e^{t}$$", "steps": [ { "type": "step", "primary": "For the non-homogeneous part $$g\\left(x\\right)=10e^{t}$$, assume a solution of the form: $$y=a_{0}e^{t}$$" }, { "type": "step", "result": "\\left(\\left(a_{0}e^{t}\\right)\\right)^{^{\\prime\\prime}}+4\\left(\\left(a_{0}e^{t}\\right)\\right)^{^{\\prime}}=10e^{t}" }, { "type": "interim", "title": "Simplify $$\\left(\\left(a_{0}e^{t}\\right)\\right)^{\\prime\\prime}+4\\left(\\left(a_{0}e^{t}\\right)\\right)^{\\prime}=10e^{t}:{\\quad}5a_{0}e^{t}=10e^{t}$$", "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": "step", "primary": "Apply the common derivative: $$\\left(e^{t}\\right)^{\\prime}=e^{t}$$", "result": "=a_{0}e^{t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a/hwLBiTUupet4un/RS9Tg27lkASVUSxyfdubUuOc4+k3hxk9aCfAWodBRxXgUex+lZb04QZg6h8t4KURUyAyGvLQ7MUdeOsalSExknVWktitLnNVCp3BjoQkf1xo/OhYtmfFK5EIB4EmGWclymbBzGl3P0Zvm2IAu+wJ2dZ6Z8=" } }, { "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": "step", "primary": "Apply the common derivative: $$\\left(e^{t}\\right)^{\\prime}=e^{t}$$", "result": "=a_{0}e^{t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a/hwLBiTUupet4un/RS9Tg27lkASVUSxyfdubUuOc4+k3hxk9aCfAWodBRxXgUex+lZb04QZg6h8t4KURUyAyGvLQ7MUdeOsalSExknVWktitLnNVCp3BjoQkf1xo/OhYtmfFK5EIB4EmGWclymbBzGl3P0Zvm2IAu+wJ2dZ6Z8=" } }, { "type": "step", "result": "=a_{0}e^{t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "a_{0}e^{t}+4\\left(\\left(a_{0}e^{t}\\right)\\right)^{^{\\prime}}=10e^{t}" }, { "type": "interim", "title": "$$\\left(\\left(a_{0}e^{t}\\right)\\right)^{\\prime}=a_{0}e^{t}$$", "input": "\\left(\\left(a_{0}e^{t}\\right)\\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": "step", "primary": "Apply the common derivative: $$\\left(e^{t}\\right)^{\\prime}=e^{t}$$", "result": "=a_{0}e^{t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L4vdMfJzYgsORgavEQNaZmlcOOTe3hj+/spmTFsuRvQcjlLRK1jUV206qo4+vRN7qX3CpoilePE+gmyoMAnyrL6UwY+EcBjsihundV8jtEJNaWyPl5p6BIMRBMKPeopAsXifnRSf0SORGb+hUg8GdypgOgMe008HXCbWZA7+wC8=" } }, { "type": "step", "result": "a_{0}e^{t}+4a_{0}e^{t}=10e^{t}" }, { "type": "step", "primary": "Simplify", "result": "5a_{0}e^{t}=10e^{t}" } ], "meta": { "interimType": "ODE Derive And Simplify 0Eq" } }, { "type": "step", "primary": "Find a solution for the coefficient(s) $$a_{0}$$" }, { "type": "interim", "title": "Solve $$5a_{0}e^{t}=10e^{t}:{\\quad}a_{0}=2$$", "input": "5a_{0}e^{t}=10e^{t}", "steps": [ { "type": "interim", "title": "Divide both sides by $$5e^{t}$$", "input": "5a_{0}e^{t}=10e^{t}", "result": "a_{0}=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$5e^{t}$$", "result": "\\frac{5a_{0}e^{t}}{5e^{t}}=\\frac{10e^{t}}{5e^{t}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{5a_{0}e^{t}}{5e^{t}}=\\frac{10e^{t}}{5e^{t}}", "result": "a_{0}=2", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{5a_{0}e^{t}}{5e^{t}}:{\\quad}a_{0}$$", "input": "\\frac{5a_{0}e^{t}}{5e^{t}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{5}{5}=1$$", "result": "=\\frac{a_{0}e^{t}}{e^{t}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{t}$$", "result": "=a_{0}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70x+vIBjQSgHdKotgK+1H4ie6jcpwOW7y06/8LHqJM9behkKrn0era9rz8TlL+x/vSImLMnZ3U1q0svyBzBjXvmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9Kfpwm/xyzEeTyD+cOZI2CQTkQHUhesSJrBCn9AIy558=" } }, { "type": "interim", "title": "Simplify $$\\frac{10e^{t}}{5e^{t}}:{\\quad}2$$", "input": "\\frac{10e^{t}}{5e^{t}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{10}{5}=2$$", "result": "=\\frac{2e^{t}}{e^{t}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{t}$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7muTaasKUxsQEDdf1VWhjRo0VaxVRrLKZU/JFmsTtgYgDnzlbPZjyKgy1eUCFsLd58vDPmi/5HbocvXjaZ5kpc03kCh3oevUunZ7/b0qFKBSBAc1PafP4ia+acEW7bvr5KvefuKOCxZsH0YVzy9lp2ImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "a_{0}=2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "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=2e^{t}" }, { "type": "step", "primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(t\\right)+4y^{\\prime}\\left(t\\right)=10e^{t}{\\quad}$$is:", "result": "y=2e^{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}+c_{2}e^{-4t}+2e^{t}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=c_{1}+c_{2}e^{-4t}+2e^{t}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "#>#ODE#>#y=c_{1}+c_{2}e^{-4t}+2e^{t}" } } }, "meta": { "showVerify": true } }