y^{''}-y=0,y(0)=0,y^'(0)=1

{ "query": { "display": "$$y^{^{\\prime\\prime}}-y=0,\\:y\\left(0\\right)=0,\\:y^{^{\\prime}}\\left(0\\right)=1$$", "symbolab_question": "ODE#y^{\\prime \\prime }-y=0,y(0)=0,y^{\\prime }(0)=1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearHomogeneous", "default": "y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(t\\right)-y=0,\\:{\\quad}y\\left(0\\right)=0,\\:{\\quad}y^{\\prime}\\left(0\\right)=1:{\\quad}y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)-y=0", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)-y=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}}-e^{γt}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}-e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}-1\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-e^{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}-e^{γt}=0" }, { "type": "step", "primary": "Factor $$e^{γt}$$", "result": "e^{γt}\\left(γ^{2}-1\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γt}\\left(γ^{2}-1\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γt}\\left(γ^{2}-1\\right)=0:{\\quad}γ=1,\\:γ=-1$$", "input": "e^{γt}\\left(γ^{2}-1\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}-1\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}-1=0$$", "result": "γ^{2}-1=0" }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "γ^{2}-1=0", "result": "γ^{2}=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "γ^{2}-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "γ^{2}=1" } ], "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{1},\\:γ=-\\sqrt{1}" }, { "type": "interim", "title": "$$\\sqrt{1}=1$$", "input": "\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "result": "=1", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KfzlHGGU7KN8vfEO0eL8NN13jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fScubCnYZOJ5L8/2gsdymw1PSOscTE6qsKVI9GkIdY/eI=" } }, { "type": "interim", "title": "$$-\\sqrt{1}=-1$$", "input": "-\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{1}=1$$", "secondary": [ "$$\\sqrt{1}=1$$" ], "result": "=-1", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWDE1Jsjy5jGSP2mctwcnCAn9lkDfZkicUGkO3EF+IpIQKToZa7Vmz9RWrIHzooCMHIu6EZfZrJ7HpyNTqg74lPlyk515FWfACaTxs0eUEM=" } }, { "type": "step", "result": "γ=1,\\:γ=-1" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=1,\\:γ=-1" }, { "type": "step", "primary": "For two real roots $$γ_{1}\\ne\\:γ_{2}$$, the general solution takes the form:$${\\quad}y=c_{1}e^{γ_{1}\\:t}+c_{2}e^{γ_{2}\\:t}$$", "result": "y=c_{1}e^{t}+c_{2}e^{-t}" }, { "type": "interim", "title": "Apply initial conditions:$${\\quad}y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}$$", "input": "y=c_{1}e^{t}+c_{2}e^{-t}", "steps": [ { "type": "step", "primary": "Plug in $$t=0$$:$${\\quad}y\\left(0\\right)=c_{1}e^{0}+c_{2}e^{-0}{\\quad}$$, and use initial condition $$y\\left(0\\right)=0$$", "result": "0=c_{1}e^{0}+c_{2}e^{-0}" }, { "type": "interim", "title": "Isolate $$c_{1}:{\\quad}c_{1}=-c_{2}$$", "input": "0=c_{1}e^{0}+c_{2}e^{-0}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "c_{1}e^{0}+c_{2}e^{-0}=0" }, { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "secondary": [ "$$e^{0}=1,\\:e^{-0}=1$$" ], "result": "1\\cdot\\:c_{1}+1\\cdot\\:c_{2}=0" }, { "type": "step", "primary": "Multiply: $$c_{1}\\cdot\\:1=c_{1}$$", "result": "c_{1}+1\\cdot\\:c_{2}=0" }, { "type": "step", "primary": "Multiply: $$c_{2}\\cdot\\:1=c_{2}$$", "result": "c_{1}+c_{2}=0" }, { "type": "interim", "title": "Move $$c_{2}\\:$$to the right side", "input": "c_{1}+c_{2}=0", "result": "c_{1}=-c_{2}", "steps": [ { "type": "step", "primary": "Subtract $$c_{2}$$ from both sides", "result": "c_{1}+c_{2}-c_{2}=0-c_{2}" }, { "type": "step", "primary": "Simplify", "result": "c_{1}=-c_{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "primary": "For $$y=c_{1}e^{t}+c_{2}e^{-t}{\\quad}$$plug in$${\\quad}c_{1}=-c_{2}$$", "result": "y=\\left(-c_{2}\\right)e^{t}+c_{2}e^{-t}" }, { "type": "step", "primary": "Simplify", "result": "y=-c_{2}e^{t}+c_{2}e^{-t}" }, { "type": "interim", "title": "Find $$y^{\\prime}\\left(t\\right):{\\quad}-c_{2}e^{t}-c_{2}e^{-t}$$", "steps": [ { "type": "interim", "title": "$$\\left(-c_{2}e^{t}+c_{2}e^{-t}\\right)^{\\prime}=-c_{2}e^{t}-c_{2}e^{-t}$$", "input": "\\left(-c_{2}e^{t}+c_{2}e^{-t}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\left(c_{2}e^{t}\\right)^{^{\\prime}}+\\left(c_{2}e^{-t}\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$\\left(c_{2}e^{t}\\right)^{\\prime}=c_{2}e^{t}$$", "input": "\\left(c_{2}e^{t}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=c_{2}\\left(e^{t}\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$\\left(e^{t}\\right)^{\\prime}=e^{t}$$", "result": "=c_{2}e^{t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ShDXALF2hfenIJ2FuGjE6g27lkASVUSxyfdubUuOc4+k3hxk9aCfAWodBRxXgUexVJaVrjms0ncpxXroATatSWvLQ7MUdeOsalSExknVWktOT1vO2AhhB//vDkwyVyPoXOZkcekNHMmq5FvdsWsrHDGl3P0Zvm2IAu+wJ2dZ6Z8=" } }, { "type": "interim", "title": "$$\\left(c_{2}e^{-t}\\right)^{\\prime}=-c_{2}e^{-t}$$", "input": "\\left(c_{2}e^{-t}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=c_{2}\\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": "=c_{2}e^{-t}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-c_{2}e^{-t}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-c_{2}e^{t}-c_{2}e^{-t}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "-c_{2}e^{t}-c_{2}e^{-t}" } ], "meta": { "interimType": "ODE Derive And Simplify 0Eq" } }, { "type": "step", "primary": "Plug in $$t=0$$:$${\\quad}y^{\\prime}\\left(0\\right)=-c_{2}e^{0}-c_{2}e^{-0}{\\quad}$$, and use initial condition $$y^{\\prime}\\left(0\\right)=1$$", "result": "1=-c_{2}e^{0}-c_{2}e^{-0}" }, { "type": "interim", "title": "Isolate $$c_{2}:{\\quad}c_{2}=-\\frac{1}{2}$$", "input": "1=-c_{2}e^{0}-c_{2}e^{-0}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-c_{2}e^{0}-c_{2}e^{-0}=1" }, { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "secondary": [ "$$e^{0}=1,\\:e^{-0}=1$$" ], "result": "-1\\cdot\\:c_{2}-1\\cdot\\:c_{2}=1" }, { "type": "step", "primary": "Add similar elements: $$-1\\cdot\\:c_{2}-1\\cdot\\:c_{2}=-2c_{2}$$", "result": "-2c_{2}=1" }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2c_{2}=1", "result": "c_{2}=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2c_{2}}{-2}=\\frac{1}{-2}" }, { "type": "step", "primary": "Simplify", "result": "c_{2}=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "primary": "For $$y=-c_{2}e^{t}+c_{2}e^{-t}{\\quad}$$plug in$${\\quad}c_{2}=-\\frac{1}{2}$$", "result": "y=-\\left(-\\frac{1}{2}\\right)e^{t}+\\left(-\\frac{1}{2}\\right)e^{-t}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}" } ], "meta": { "interimType": "Plug In Initial Condition 0Eq" } }, { "type": "step", "result": "y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "#>#ODE#>#y=\\frac{1}{2}e^{t}-\\frac{1}{2}e^{-t}" } } }, "meta": { "showVerify": true } }