y^{''}-4y=8e^{2x}

{ "query": { "display": "$$y^{^{\\prime\\prime}}-4y=8e^{2x}$$", "symbolab_question": "ODE#y^{\\prime \\prime }-4y=8e^{2x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "ConstCoeffLinearNonHomogeneous", "default": "y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$y^{\\prime\\prime}\\left(x\\right)-4y=8e^{2x}:{\\quad}y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}$$", "input": "y^{\\prime\\prime}\\left(x\\right)-4y=8e^{2x}", "steps": [ { "type": "interim", "title": "Solve linear ODE:$${\\quad}y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}$$", "input": "y^{\\prime\\prime}\\left(x\\right)-4y=8e^{2x}", "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 $$y^{\\prime\\prime}\\left(x\\right)-4y=0:{\\quad}y=c_{1}e^{2x}+c_{2}e^{-2x}$$", "input": "y^{\\prime\\prime}\\left(x\\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 $$e^{γx}$$", "secondary": [ "Rewrite the equation with $$y=e^{γx}$$" ], "result": "\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime\\prime}}-4e^{γx}=0" }, { "type": "interim", "title": "Simplify $$\\left(\\left(e^{γx}\\right)\\right)^{\\prime\\prime}-4e^{γx}=0:{\\quad}e^{γx}\\left(γ^{2}-4\\right)=0$$", "steps": [ { "type": "step", "result": "\\left(\\left(e^{γx}\\right)\\right)^{^{\\prime\\prime}}-4e^{γx}=0" }, { "type": "interim", "title": "$$\\left(e^{γx}\\right)^{\\prime\\prime}=γ^{2}e^{γx}$$", "input": "\\left(e^{γx}\\right)^{\\prime\\prime}", "steps": [ { "type": "interim", "title": "$$\\left(e^{γx}\\right)^{\\prime}=e^{γx}γ$$", "input": "\\left(e^{γx}\\right)^{\\prime}", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{γx}\\left(γx\\right)^{\\prime}$$", "input": "\\left(e^{γx}\\right)^{\\prime}", "result": "=e^{γx}\\left(γx\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=γx$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γx\\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(γx\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=γx$$", "result": "=e^{γx}\\left(γx\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcBDY8sG8wkKf3KVMoxpWVQssjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5M1thhQ/Ed/UbgOWE3OQBzQDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$\\left(γx\\right)^{\\prime}=γ$$", "input": "\\left(γx\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=γx^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=γ\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=γ", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EVB+jU4LMswLFUscsVdfVSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tV6EkZL7Rz6t72e1SEoLgfPvRCDs4D3rcIVpx7C72k9c" } }, { "type": "step", "result": "=e^{γx}γ" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\left(e^{γx}γ\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$\\left(e^{γx}γ\\right)^{\\prime}=γ^{2}e^{γx}$$", "input": "\\left(e^{γx}γ\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=γ\\left(e^{γx}\\right)^{^{\\prime}}" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{γx}\\left(γx\\right)^{\\prime}$$", "input": "\\left(e^{γx}\\right)^{\\prime}", "result": "=e^{γx}\\left(γx\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=γx$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γx\\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(γx\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=γx$$", "result": "=e^{γx}\\left(γx\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcBDY8sG8wkKf3KVMoxpWVQssjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5M1thhQ/Ed/UbgOWE3OQBzQDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$\\left(γx\\right)^{\\prime}=γ$$", "input": "\\left(γx\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=γx^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=γ\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=γ", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EVB+jU4LMswLFUscsVdfVSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tV6EkZL7Rz6t72e1SEoLgfPvRCDs4D3rcIVpx7C72k9c" } }, { "type": "step", "result": "=γe^{γx}γ" }, { "type": "interim", "title": "Simplify $$γe^{γx}γ:{\\quad}γ^{2}e^{γx}$$", "input": "γe^{γx}γ", "result": "=γ^{2}e^{γx}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$γγ=\\:γ^{1+1}$$" ], "result": "=e^{γx}γ^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=e^{γx}γ^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ogc+QG1cya0zQe2NAtljy96GQqufR6tr2vPxOUv7H++vcvW4o70lTWJV7TqReTsbmx4rCXhbsN+br+uOYP22UU3kCh3oevUunZ7/b0qFKBStCRMtul5SOs/SBwPTbaWuzHQDH6pWoNP3G07HzBhaLA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=γ^{2}e^{γx}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "γ^{2}e^{γx}-4e^{γx}=0" }, { "type": "step", "primary": "Factor $$e^{γx}$$", "result": "e^{γx}\\left(γ^{2}-4\\right)=0" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "e^{γx}\\left(γ^{2}-4\\right)=0" }, { "type": "interim", "title": "Solve $$e^{γx}\\left(γ^{2}-4\\right)=0:{\\quad}γ=2,\\:γ=-2$$", "input": "e^{γx}\\left(γ^{2}-4\\right)=0", "steps": [ { "type": "step", "primary": "Since $$e^{γx}\\ne\\:0$$, solving $$e^{γx}\\left(γ^{2}-4\\right)=0$$<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": "Add $$4$$ to 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": "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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{4},\\:γ=-\\sqrt{4}" }, { "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": "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VpfTWfsX6bgJBXQ1iIjjMSAn9lkDfZkicUGkO3EF+Irv9b+CB9cfiKbG2PM3RjXgMHIu6EZfZrJ7HpyNTqg74q7TwYnH2OBc9PfvZ6CbknQ=" } }, { "type": "step", "result": "γ=2,\\:γ=-2" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "γ=2,\\:γ=-2" }, { "type": "step", "primary": "For two real roots $$γ_{1}\\ne\\:γ_{2}$$, the general solution takes the form:$${\\quad}y=c_{1}e^{γ_{1}\\:x}+c_{2}e^{γ_{2}\\:x}$$", "result": "y=c_{1}e^{2x}+c_{2}e^{-2x}" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Find By Solving Title 2Eq" } }, { "type": "interim", "title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(x\\right)-4y=8e^{2x}:{\\quad}y=2xe^{2x}$$", "steps": [ { "type": "step", "primary": "For the non-homogeneous part $$g\\left(x\\right)=8e^{2x}$$, assume a solution of the form: $$y=a_{0}xe^{2x}$$" }, { "type": "step", "result": "\\left(\\left(a_{0}xe^{2x}\\right)\\right)^{^{\\prime\\prime}}-4a_{0}xe^{2x}=8e^{2x}" }, { "type": "interim", "title": "Simplify $$\\left(\\left(a_{0}xe^{2x}\\right)\\right)^{\\prime\\prime}-4a_{0}xe^{2x}=8e^{2x}:{\\quad}4a_{0}e^{2x}=8e^{2x}$$", "steps": [ { "type": "interim", "title": "$$\\left(\\left(a_{0}xe^{2x}\\right)\\right)^{\\prime\\prime}=a_{0}\\left(4e^{2x}x+4e^{2x}\\right)$$", "input": "\\left(\\left(a_{0}xe^{2x}\\right)\\right)^{\\prime\\prime}", "steps": [ { "type": "interim", "title": "$$\\left(a_{0}xe^{2x}\\right)^{\\prime}=a_{0}\\left(e^{2x}+2e^{2x}x\\right)$$", "input": "\\left(a_{0}xe^{2x}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\left(xe^{2x}\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=e^{2x}$$" ], "result": "=a_{0}\\left(x^{^{\\prime}}e^{2x}+\\left(e^{2x}\\right)^{^{\\prime}}x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$x^{\\prime}=1$$", "input": "x^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PwIewkjm/nXCEo0Ji146t6boRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wrOr42DoRa4lmvtW+XTBjyKea8SiFhbPG8yoKCXYENvrw=" } }, { "type": "interim", "title": "$$\\left(e^{2x}\\right)^{\\prime}=e^{2x}\\cdot\\:2$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{2x}\\left(2x\\right)^{\\prime}$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "result": "=e^{2x}\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=2x$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(2x\\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(2x\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=e^{2x}\\left(2x\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71GjX8/WOlVQoR3720Yfm9aRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOcTTG1MQjF4JOCdGL5fDxZfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(2x\\right)^{\\prime}=2$$", "input": "\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2x^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iEluvAhAB6qVseHmXOoGm8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMolkIAcH8DCZe9Bzo2oZbtQQDuUeqnBwgbzjs2dJUq2K" } }, { "type": "step", "result": "=e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=a_{0}\\left(1\\cdot\\:e^{2x}+e^{2x}\\cdot\\:2x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=a_{0}\\left(e^{2x}+2e^{2x}x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\left(a_{0}\\left(e^{2x}+2e^{2x}x\\right)\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$\\left(a_{0}\\left(e^{2x}+2e^{2x}x\\right)\\right)^{\\prime}=a_{0}\\left(4e^{2x}x+4e^{2x}\\right)$$", "input": "\\left(a_{0}\\left(e^{2x}+2e^{2x}x\\right)\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=a_{0}\\left(e^{2x}+2e^{2x}x\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=a_{0}\\left(\\left(e^{2x}\\right)^{^{\\prime}}+\\left(2e^{2x}x\\right)^{^{\\prime}}\\right)" }, { "type": "interim", "title": "$$\\left(e^{2x}\\right)^{\\prime}=e^{2x}\\cdot\\:2$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{2x}\\left(2x\\right)^{\\prime}$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "result": "=e^{2x}\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=2x$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(2x\\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(2x\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=e^{2x}\\left(2x\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71GjX8/WOlVQoR3720Yfm9aRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOcTTG1MQjF4JOCdGL5fDxZfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(2x\\right)^{\\prime}=2$$", "input": "\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2x^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iEluvAhAB6qVseHmXOoGm8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMolkIAcH8DCZe9Bzo2oZbtQQDuUeqnBwgbzjs2dJUq2K" } }, { "type": "step", "result": "=e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\left(2e^{2x}x\\right)^{\\prime}=2\\left(2e^{2x}x+e^{2x}\\right)$$", "input": "\\left(2e^{2x}x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\left(e^{2x}x\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{2x},\\:g=x$$" ], "result": "=2\\left(\\left(e^{2x}\\right)^{^{\\prime}}x+x^{^{\\prime}}e^{2x}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\left(e^{2x}\\right)^{\\prime}=e^{2x}\\cdot\\:2$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{2x}\\left(2x\\right)^{\\prime}$$", "input": "\\left(e^{2x}\\right)^{\\prime}", "result": "=e^{2x}\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=2x$$" ], "result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(2x\\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(2x\\right)^{^{\\prime}}" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=e^{2x}\\left(2x\\right)^{^{\\prime}}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71GjX8/WOlVQoR3720Yfm9aRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUn04noP5Jb9NWXDshKwxOQOcTTG1MQjF4JOCdGL5fDxZfH6kVvau+ENR7awhAU+mEsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\left(2x\\right)^{\\prime}=2$$", "input": "\\left(2x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2x^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iEluvAhAB6qVseHmXOoGm8PQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMolkIAcH8DCZe9Bzo2oZbtQQDuUeqnBwgbzjs2dJUq2K" } }, { "type": "step", "result": "=e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$x^{\\prime}=1$$", "input": "x^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PwIewkjm/nXCEo0Ji146t6boRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wrOr42DoRa4lmvtW+XTBjyKea8SiFhbPG8yoKCXYENvrw=" } }, { "type": "step", "result": "=2\\left(e^{2x}\\cdot\\:2x+1\\cdot\\:e^{2x}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=2\\left(2e^{2x}x+e^{2x}\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=a_{0}\\left(e^{2x}\\cdot\\:2+2\\left(2e^{2x}x+e^{2x}\\right)\\right)" }, { "type": "interim", "title": "Expand $$e^{2x}\\cdot\\:2+2\\left(2e^{2x}x+e^{2x}\\right):{\\quad}4e^{2x}x+4e^{2x}$$", "input": "e^{2x}\\cdot\\:2+2\\left(2e^{2x}x+e^{2x}\\right)", "result": "=a_{0}\\left(4e^{2x}x+4e^{2x}\\right)", "steps": [ { "type": "step", "result": "=2e^{2x}+2\\left(2e^{2x}x+e^{2x}\\right)" }, { "type": "interim", "title": "Expand $$2\\left(2e^{2x}x+e^{2x}\\right):{\\quad}4e^{2x}x+2e^{2x}$$", "input": "2\\left(2e^{2x}x+e^{2x}\\right)", "result": "=e^{2x}\\cdot\\:2+4e^{2x}x+2e^{2x}", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=2e^{2x}x,\\:c=e^{2x}$$" ], "result": "=2\\cdot\\:2e^{2x}x+2e^{2x}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4e^{2x}x+2e^{2x}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qExtqw+qFkOxHY4+cTIncvxm3cSsufIZ9Cd+pJShZK/MwViaLUXkeD+JukROhWdjcXWocQe9GvpE0N1OKNhnMP8//6/nV5O4fb8Xgwi7mar8bYA0b6V2RSTOZ7Os9NODUKSN5OF+PnZIDwbtsKpq47dz+v04qfGRqtyNtJnTjYc=" } }, { "type": "interim", "title": "Simplify $$e^{2x}\\cdot\\:2+4e^{2x}x+2e^{2x}:{\\quad}4e^{2x}x+4e^{2x}$$", "input": "e^{2x}\\cdot\\:2+4e^{2x}x+2e^{2x}", "result": "=4e^{2x}x+4e^{2x}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=4e^{2x}x+2e^{2x}+2e^{2x}" }, { "type": "step", "primary": "Add similar elements: $$2e^{2x}+2e^{2x}=4e^{2x}$$", "result": "=4e^{2x}x+4e^{2x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Py+gJJkxR4xZjJCXEm8tVyvchxZQh7HlmNwIrk+5GczdE6r+4mYZTuwaWBOmnB4Oo5FYteSPKwXny4uCMrdsK2OF17TayurCKNKfpYif+qI/y9DKGIPglJ+qMi9xDu2Kq+GZ7j6klu8hDcbJq5wtQQQNrqHpjiSUwwKlVZ7jxTPHOJLun1Ln4Bb1yYC4MVfGv74GQiLapk2I7yZErOiqLw==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=a_{0}\\left(4e^{2x}x+4e^{2x}\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "a_{0}\\left(4e^{2x}x+4e^{2x}\\right)-4a_{0}xe^{2x}=8e^{2x}" }, { "type": "step", "primary": "Simplify", "result": "4a_{0}e^{2x}=8e^{2x}" } ], "meta": { "interimType": "ODE Derive And Simplify 0Eq" } }, { "type": "step", "primary": "Find a solution for the coefficient(s) $$a_{0}$$" }, { "type": "interim", "title": "Solve $$4a_{0}e^{2x}=8e^{2x}:{\\quad}a_{0}=2$$", "input": "4a_{0}e^{2x}=8e^{2x}", "steps": [ { "type": "interim", "title": "Divide both sides by $$4e^{2x}$$", "input": "4a_{0}e^{2x}=8e^{2x}", "result": "a_{0}=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$4e^{2x}$$", "result": "\\frac{4a_{0}e^{2x}}{4e^{2x}}=\\frac{8e^{2x}}{4e^{2x}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{4a_{0}e^{2x}}{4e^{2x}}=\\frac{8e^{2x}}{4e^{2x}}", "result": "a_{0}=2", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4a_{0}e^{2x}}{4e^{2x}}:{\\quad}a_{0}$$", "input": "\\frac{4a_{0}e^{2x}}{4e^{2x}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{4}=1$$", "result": "=\\frac{a_{0}e^{2x}}{e^{2x}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{2x}$$", "result": "=a_{0}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MCwKDkJBuYSfTqKHPRP9t4tdw07JDSxcXhWNRgaTm4gtOtZYwUjyXhDTsNnn6Elrf64AHx7R+XXMwtlqucIhS/8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl/CYE6PmcSjhPn7jMMubxSjNmwtZuiTnHLgYDSyr4VHGJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Simplify $$\\frac{8e^{2x}}{4e^{2x}}:{\\quad}2$$", "input": "\\frac{8e^{2x}}{4e^{2x}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=\\frac{2e^{2x}}{e^{2x}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{2x}$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OBjb1HUOy0s12u8/jfYlsO1T6OLAgt22eJX9P38/u+7NGoPE9TME3q+OPmgkv2RQ3tPsgY8+I795OPEaQ1ATH3ql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HzsypQYeYg4D8n+sXX+5CGVtl11sFEBowxkl8AtNH9Gs=" } }, { "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 $$y=a_{0}xe^{2x}$$", "result": "y=2xe^{2x}" }, { "type": "step", "primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(x\\right)-4y=8e^{2x}{\\quad}$$is:", "result": "y=2xe^{2x}" } ], "meta": { "interimType": "Generic Find That Satisfies Title 2Eq" } }, { "type": "step", "primary": "The general solution $$y=y_h+y_p$$ is:", "result": "y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}" } ], "meta": { "interimType": "ODE Solve Linear 0Eq" } }, { "type": "step", "result": "y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=c_{1}e^{2x}+c_{2}e^{-2x}+2xe^{2x}" } } }, "meta": { "showVerify": true } }