y^{''}-21y^'+108y=0

{ "query": { "display": "$$y^{^{\\prime\\prime}}-21y^{^{\\prime}}+108y=0$$", "symbolab_question": "ODE#y^{\\prime \\prime }-21y^{\\prime }+108y=0" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearHomogeneous", "default": "y=c_{1}e^{12t}+c_{2}e^{9t}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(t\\right)-21y^{\\prime}\\left(t\\right)+108y=0:{\\quad}y=c_{1}e^{12t}+c_{2}e^{9t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)-21y^{\\prime}\\left(t\\right)+108y=0", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=c_{1}e^{12t}+c_{2}e^{9t}$$", "input": "y^{\\prime\\prime}\\left(t\\right)-21y^{\\prime}\\left(t\\right)+108y=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}}-21\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+108e^{γt}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}-21\\left(\\left(e^{γt}\\right)\\right)^{\\prime}+108e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}-21γ+108\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-21\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+108e^{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}-21\\left(e^{γt}\\right)^{^{\\prime}}+108e^{γt}=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}-21e^{γt}γ+108e^{γt}=0" }, { "type": "step", "primary": "Factor $$e^{γt}$$", "result": "e^{γt}\\left(γ^{2}-21γ+108\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γt}\\left(γ^{2}-21γ+108\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γt}\\left(γ^{2}-21γ+108\\right)=0:{\\quad}γ=12,\\:γ=9$$", "input": "e^{γt}\\left(γ^{2}-21γ+108\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}-21γ+108\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}-21γ+108=0$$", "result": "γ^{2}-21γ+108=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "γ^{2}-21γ+108=0", "result": "{γ}_{1,\\:2}=\\frac{-\\left(-21\\right)\\pm\\:\\sqrt{\\left(-21\\right)^{2}-4\\cdot\\:1\\cdot\\:108}}{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 $${\\quad}a=1,\\:b=-21,\\:c=108$$", "result": "{γ}_{1,\\:2}=\\frac{-\\left(-21\\right)\\pm\\:\\sqrt{\\left(-21\\right)^{2}-4\\cdot\\:1\\cdot\\:108}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-21\\right)^{2}-4\\cdot\\:1\\cdot\\:108}=3$$", "input": "\\sqrt{\\left(-21\\right)^{2}-4\\cdot\\:1\\cdot\\:108}", "result": "{γ}_{1,\\:2}=\\frac{-\\left(-21\\right)\\pm\\:3}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-21\\right)^{2}=21^{2}$$" ], "result": "=\\sqrt{21^{2}-4\\cdot\\:1\\cdot\\:108}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:108=432$$", "result": "=\\sqrt{21^{2}-432}" }, { "type": "step", "primary": "$$21^{2}=441$$", "result": "=\\sqrt{441-432}" }, { "type": "step", "primary": "Subtract the numbers: $$441-432=9$$", "result": "=\\sqrt{9}" }, { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wNuCBo4dxpnbsuoE8X0LykFj7CiO8FQNDSt2Nz1mnF4U/8BohnPGUT377bP1We++CUCWbkwGOY7PqKo3U/JLJcAPqbrk7jyO14RRueLYbUGxGBnOQAMXlURTU9qIbJVRIzZYM9Pi8AugCP1qTu5CEZbTdMXOMZerv9zM4JVulj4=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{γ}_{1}=\\frac{-\\left(-21\\right)+3}{2\\cdot\\:1},\\:{γ}_{2}=\\frac{-\\left(-21\\right)-3}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$γ=\\frac{-\\left(-21\\right)+3}{2\\cdot\\:1}:{\\quad}12$$", "input": "\\frac{-\\left(-21\\right)+3}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{21+3}{2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$21+3=24$$", "result": "=\\frac{24}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{24}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{24}{2}=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Te2vGme0vNwGYVHmHXimLsZOo4D+uTEXY33RTXF+DcPehkKrn0era9rz8TlL+x/vlsKh9clUY1HBeZ58AZwjeKkhYmpKOTSDkevtNXkcJueO2P+T10YKHNT0ittNHCFTADyQWdqYynpsnR8jROuTkg==" } }, { "type": "interim", "title": "$$γ=\\frac{-\\left(-21\\right)-3}{2\\cdot\\:1}:{\\quad}9$$", "input": "\\frac{-\\left(-21\\right)-3}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{21-3}{2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$21-3=18$$", "result": "=\\frac{18}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{18}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{18}{2}=9$$", "result": "=9" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7W4dQKfSLChVbhuEr2kbMocZOo4D+uTEXY33RTXF+DcPehkKrn0era9rz8TlL+x/vPelnHf6ZVeObZvlubBF5IjpcHX7bef9LI9MEYMv1Vmr4kDckkdtX7+judTWIgBD6DtGC/wTvnB+aHeEUayusew==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "γ=12,\\:γ=9" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=12,\\:γ=9" }, { "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^{12t}+c_{2}e^{9t}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=c_{1}e^{12t}+c_{2}e^{9t}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "#>#ODE#>#y=c_{1}e^{12t}+c_{2}e^{9t}" } } }, "meta": { "showVerify": true } }