(dy)/(dx)+8y=x^2y^2

{ "query": { "display": "$$\\frac{dy}{dx}+8y=x^{2}y^{2}$$", "symbolab_question": "ODE#\\frac{dy}{dx}+8y=x^{2}y^{2}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstBernoulli", "default": "y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{dy}{dx}+8y=x^{2}y^{2}:{\\quad}y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}$$", "input": "\\frac{dy}{dx}+8y=x^{2}y^{2}", "steps": [ { "type": "interim", "title": "Solve Bernoulli ODE:$${\\quad}y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}$$", "input": "\\frac{dy}{dx}+8y=x^{2}y^{2}", "steps": [ { "type": "definition", "title": "First order Bernoulli Ordinary Differential Equation", "text": "A first order Bernoulli ODE has the form of $$y'+p\\left(x\\right)y=q\\left(x\\right)y^n$$" }, { "type": "step", "primary": "Substitute $$\\frac{dy}{dx}$$ with $$y^{\\prime}\\left(x\\right)$$", "result": "y^{^{\\prime}}\\left(x\\right)+8y=x^{2}y^{2}" }, { "type": "interim", "title": "The equation is in first order Bernoulli ODE form", "steps": [ { "type": "step", "primary": "$$y'+p\\left(x\\right)y=q\\left(x\\right)y^n$$", "secondary": [ "$$p\\left(x\\right)=8,\\:{\\quad}q\\left(x\\right)=x^{2},\\:{\\quad}n=2$$" ] } ], "meta": { "interimType": "Already Canon First Order ODE Bernoulli 2Eq" } }, { "type": "step", "primary": "The general solution is obtained by substituting $$v=y^{1-n}\\:$$and solving $$\\frac{1}{1-n}v'+p\\left(x\\right)v=q\\left(x\\right)$$" }, { "type": "interim", "title": "Transform to $$\\frac{1}{1-n}v'+p\\left(x\\right)v=q\\left(x\\right):{\\quad}-ν^{\\prime}+8ν=x^{2}$$", "input": "y^{\\prime}\\left(x\\right)+8y=x^{2}y^{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$y^{2}$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}+\\frac{8y}{y^{2}}=\\frac{x^{2}y^{2}}{y^{2}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}+\\frac{8}{y}=x^{2}" }, { "type": "step", "primary": "If $$\\nu=y^{1-n}\\:$$then $$\\nu'=\\left(1-n\\right)\\frac{y'}{y^n}.{\\quad}$$Therefore, the equation can be transformed to a linear ODE of $$\\nu$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}+\\frac{8}{y}=x^{2}" }, { "type": "step", "primary": "Substitute $$y^{-1}$$ with $$ν$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}+8ν=x^{2}" }, { "type": "interim", "title": "Compute $$ν^{\\prime}:{\\quad}-\\frac{y^{\\prime}\\left(x\\right)}{y^{2}}$$", "input": "\\left(y^{-1}\\right)^{\\prime}", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\frac{1}{y^{2}}y^{\\prime}\\left(x\\right)$$", "input": "\\left(y^{-1}\\right)^{\\prime}", "result": "=-\\frac{1}{y^{2}}y^{\\prime}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{-1},\\:\\:u=y$$" ], "result": "=\\left(u^{-1}\\right)^{^{\\prime}}y^{^{\\prime}}\\left(x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\left(u^{-1}\\right)^{\\prime}=-\\frac{1}{u^{2}}$$", "input": "\\left(u^{-1}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=-1\\cdot\\:u^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:u^{-1-1}:{\\quad}-\\frac{1}{u^{2}}$$", "input": "-1\\cdot\\:u^{-1-1}", "result": "=-\\frac{1}{u^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:u^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-2}=\\frac{1}{u^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{u^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{u^{2}}=\\frac{1}{u^{2}}$$", "result": "=-\\frac{1}{u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CXvK7JUUE3qLvMjBzSjK6qiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrYtPZxRVru5JdAr5eMQ8tHf8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlqk5ccqytIBl7J4uu5P3aKRF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\frac{1}{u^{2}}y^{^{\\prime}}\\left(x\\right)" }, { "type": "step", "primary": "Substitute back $$u=y$$", "result": "=-\\frac{1}{y^{2}}y^{^{\\prime}}\\left(x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bzJm943l4v1Ua2yvAp8/cqRMrCnJ6xySUWC3ZSXynKYHjZ0JmeAC3ZSEmMxWRNYu6XlRsqVFSAmW95ptSxLnUhFATDJ10spCLYB2r5ccvBfc+vSZ1BnD9Ail6zE+vfu+NVP7bINoMrRyz79fYTgxjoEFMST8lDZxn1Yq5HMKVTsbNyiL8Z1p0A31j/ogQw2wq7BvZX0JQ7a9rG/wceXSdQ==" } }, { "type": "interim", "title": "Simplify $$-\\frac{1}{y^{2}}y^{\\prime}\\left(x\\right):{\\quad}-\\frac{y^{\\prime}\\left(x\\right)}{y^{2}}$$", "input": "-\\frac{1}{y^{2}}y^{\\prime}\\left(x\\right)", "result": "=-\\frac{y^{\\prime}\\left(x\\right)}{y^{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1y^{^{\\prime}}\\left(x\\right)}{y^{2}}" }, { "type": "step", "primary": "Multiply: $$1y^{\\prime}\\left(x\\right)=y^{\\prime}\\left(x\\right)$$", "result": "=-\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RWs9oh740CnOGobHjga0tr9/Mf/38ngQdLG4TBlnbNwtOtZYwUjyXhDTsNnn6ElryOsg4xTbsj8PJfnagYu7Q/oSQ1vU7I9xsOwsLjcgwP5u2I3yuSHGyg64ARnL7+ppeqXxdc+rps1CUyb7fqI2GbjO0VoSw41Ks7aVPvS9CM6GrVm+wrCK4jwheK9hER8PCHMiQeIFtc/2vus1B8+XPg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Generic Compute Title 1Eq" } }, { "type": "step", "primary": "Substitute $$-\\frac{y^{\\prime}\\left(x\\right)}{y^{2}}$$ with $$ν^{\\prime}$$", "result": "-ν^{^{\\prime}}+8ν=x^{2}" } ], "meta": { "interimType": "Transform To Bernoulli 1Eq" } }, { "type": "interim", "title": "Solve $$-ν^{\\prime}+8ν=x^{2}:{\\quad}ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}$$", "input": "-ν^{\\prime}+8ν=x^{2}", "steps": [ { "type": "definition", "title": "First order linear Ordinary Differential Equation", "text": "A first order linear ODE has the form of $$y'\\left(x\\right)+p\\left(x\\right)y=q\\left(x\\right)$$" }, { "type": "interim", "title": "Rewrite in the form of a first order linear ODE", "input": "-ν^{\\prime}+8ν=x^{2}", "result": "ν^{\\prime}-8ν=-x^{2}", "steps": [ { "type": "step", "primary": "Standard form of a first order linear ODE:", "secondary": [ "$$y'\\left(x\\right)+p\\left(x\\right){\\cdot}y=q\\left(x\\right)$$" ] }, { "type": "step", "result": "-ν^{^{\\prime}}+8ν=x^{2}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "-\\frac{ν^{^{\\prime}}}{-1}+\\frac{8ν}{-1}=\\frac{x^{2}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "ν^{^{\\prime}}-8ν=-x^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$p\\left(x\\right)=-8,\\:{\\quad}q\\left(x\\right)=-x^{2}$$" ], "result": "ν^{^{\\prime}}-8ν=-x^{2}" } ], "meta": { "interimType": "Canon First Order ODE 2Eq" } }, { "type": "interim", "title": "Find the integration factor:$${\\quad}μ\\left(x\\right)=e^{-8x}$$", "steps": [ { "type": "step", "primary": "Find the integrating factor $$\\mu\\left(x\\right)$$, so that: $$\\mu\\left(x\\right){\\cdot}p\\left(x\\right)=\\mu'\\left(x\\right)$$", "result": "μ^{^{\\prime}}\\left(x\\right)=μ\\left(x\\right)p\\left(x\\right)" }, { "type": "step", "primary": "Divide both sides by $$μ\\left(x\\right)$$", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=\\frac{μ\\left(x\\right)p\\left(x\\right)}{μ\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{μ^{^{\\prime}}\\left(x\\right)}{μ\\left(x\\right)}=p\\left(x\\right)" }, { "type": "step", "primary": "$$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=\\frac{μ^{\\prime}\\left(x\\right)}{μ\\left(x\\right)}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=p\\left(x\\right)", "meta": { "general_rule": { "extension": "$$\\frac{d}{dx}\\left(\\ln\\left(f\\left(x\\right)\\right)\\right)=\\frac{\\frac{d}{dx}f\\left(x\\right)}{f\\left(x\\right)}$$" } } }, { "type": "step", "primary": "$$p\\left(x\\right)=-8$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=-8" }, { "type": "interim", "title": "Solve $$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-8:{\\quad}μ\\left(x\\right)=e^{-8x+c_{1}}$$", "input": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=-8", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "\\ln\\left(μ\\left(x\\right)\\right)=\\int\\:-8dx" }, { "type": "interim", "title": "$$\\int\\:-8dx=-8x+c_{1}$$", "input": "\\int\\:-8dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=\\left(-8\\right)x" }, { "type": "step", "primary": "Simplify", "result": "=-8x", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-8x+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "\\ln\\left(μ\\left(x\\right)\\right)=-8x+c_{1}" }, { "type": "interim", "title": "Isolate $$μ\\left(x\\right):{\\quad}μ\\left(x\\right)=e^{-8x+c_{1}}$$", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-8x+c_{1}", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(μ\\left(x\\right)\\right)=-8x+c_{1}", "result": "μ\\left(x\\right)=e^{-8x+c_{1}}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(μ\\left(x\\right)\\right)=-8x+c_{1}\\quad\\:\\Rightarrow\\:\\quad\\:μ\\left(x\\right)=e^{-8x+c_{1}}$$" ], "result": "μ\\left(x\\right)=e^{-8x+c_{1}}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xZ/kvw9iKo4JlZMns/8rf2leN4hDoej7/uNdJZ3VF8852FciCV6Q/ZuTzBHIPdDywh1dx4QxVQTFFuRGwejJdxBZfaSS8J7oFLak19JT8wN0e2pmA0w6UnyCMIIVIIDtI/YXAsnx9sPnn2MeOn7NelIG/qLOF4tVQpD96aiQBTLNaBe9sIstZ/0G32TTK32u" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=e^{-8x+c_{1}}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=e^{-8x+c_{1}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$e^{-8x+c_{1}}=e^{-8x}e^{c_{1}}$$" ], "result": "μ\\left(x\\right)=e^{-8x}e^{c_{1}}" }, { "type": "step", "primary": "The constant $$e^{c_{1}}\\:$$can be dropped (it will be absorbed into C)", "result": "μ\\left(x\\right)=e^{-8x}" } ], "meta": { "interimType": "Integrating Factor Top 0Eq" } }, { "type": "interim", "title": "Put the equation in the form $$\\left(\\mu\\left(x\\right){\\cdot}y\\right)'=\\mu\\left(x\\right){\\cdot}q\\left(x\\right):{\\quad}\\left(e^{-8x}ν\\right)^{\\prime}=-e^{-8x}x^{2}$$", "steps": [ { "type": "step", "primary": "Multiply by the integration factor, $$\\mu\\left(x\\right)$$ and rewrtie the equation as<br/>$$\\left(\\mu\\left(x\\right)\\cdot\\:y\\left(x\\right)\\right)'=\\mu\\:\\left(x\\right)\\cdot\\:q\\left(x\\right)$$", "result": "ν^{^{\\prime}}-8ν=-x^{2}" }, { "type": "step", "primary": "Multiply both sides by the integrating factor, $$e^{-8x}$$", "result": "ν^{^{\\prime}}e^{-8x}-8νe^{-8x}=\\left(-x^{2}\\right)e^{-8x}" }, { "type": "step", "primary": "Simplify", "result": "ν^{^{\\prime}}e^{-8x}-8νe^{-8x}=-e^{-8x}x^{2}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{-8x},\\:g=ν:{\\quad}ν^{\\prime}e^{-8x}-8νe^{-8x}=\\left(e^{-8x}ν\\right)^{\\prime}$$" ], "result": "\\left(e^{-8x}ν\\right)^{^{\\prime}}=-e^{-8x}x^{2}" } ], "meta": { "interimType": "Bring Linear To Derivative Form Left 0Eq" } }, { "type": "interim", "title": "Solve $$\\left(e^{-8x}ν\\right)^{\\prime}=-e^{-8x}x^{2}:{\\quad}ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}$$", "input": "\\left(e^{-8x}ν\\right)^{\\prime}=-e^{-8x}x^{2}", "steps": [ { "type": "step", "primary": "If$${\\quad}f^{^{\\prime}}\\left(x\\right)=g\\left(x\\right){\\quad}$$then$${\\quad}f\\left(x\\right)=\\int{g\\left(x\\right)}dx$$", "result": "e^{-8x}ν=\\int\\:-e^{-8x}x^{2}dx" }, { "type": "interim", "title": "$$\\int\\:-e^{-8x}x^{2}dx=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)+c_{1}$$", "input": "\\int\\:-e^{-8x}x^{2}dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\int\\:e^{-8x}x^{2}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:e^{-8x}x^{2}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=-8x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-8$$", "input": "\\left(-8x\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-8x^{^{\\prime}}" }, { "type": "step", "primary": "Apply the common derivative: $$x^{\\prime}=1$$", "result": "=-8\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=-8", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X9APncSnzcq77CXuhlIFwSENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXocFAGz9x8UkrvPSx7u5WrDvqIOlxNXEONDm3M0PlIv9rQ4P9owGPjDfb/Z7hC644Q" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-8dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{1}{8}\\right)du$$" }, { "type": "step", "result": "=\\int\\:e^{u}x^{2}\\left(-\\frac{1}{8}\\right)du" }, { "type": "step", "result": "=\\int\\:-\\frac{1}{8}e^{u}x^{2}du" }, { "type": "interim", "title": "$$u=-8x\\quad\\Rightarrow\\quad\\:x=-\\frac{u}{8}$$", "input": "-8x=u", "steps": [ { "type": "interim", "title": "Divide both sides by $$-8$$", "input": "-8x=u", "result": "x=-\\frac{u}{8}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-8$$", "result": "\\frac{-8x}{-8}=\\frac{u}{-8}" }, { "type": "step", "primary": "Simplify", "result": "x=-\\frac{u}{8}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{8}e^{u}\\left(-\\frac{u}{8}\\right)^{2}du" }, { "type": "interim", "title": "Simplify $$-\\frac{1}{8}e^{u}\\left(-\\frac{u}{8}\\right)^{2}:{\\quad}-\\frac{e^{u}u^{2}}{512}$$", "input": "-\\frac{1}{8}e^{u}\\left(-\\frac{u}{8}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{u}{8}\\right)^{2}=\\frac{u^{2}}{8^{2}}$$", "input": "\\left(-\\frac{u}{8}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\frac{u}{8}\\right)^{2}=\\left(\\frac{u}{8}\\right)^{2}$$" ], "result": "=\\left(\\frac{u}{8}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{u^{2}}{8^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VbpMPLL9RCOevvaSlFUkiOiEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPI+SACyqIgbZi9g2UPIgegpcoQ7VgSFZ7F7jg3rh7wERN85njq1sjviV873seyarpejGG1CVnt+aCWPkrr7LQH+yndTm1BHBCOGxCrr1vgiuskt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-\\frac{1}{8}e^{u}\\frac{u^{2}}{8^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=-\\frac{1\\cdot\\:u^{2}e^{u}}{8\\cdot\\:8^{2}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:u^{2}=u^{2}$$", "result": "=-\\frac{e^{u}u^{2}}{8^{2}\\cdot\\:8}" }, { "type": "interim", "title": "$$8\\cdot\\:8^{2}=8^{3}$$", "input": "8\\cdot\\:8^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$8\\cdot\\:8^{2}=\\:8^{1+2}$$" ], "result": "=8^{1+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=8^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sA4yNM0mqAllmW/CUheQyC061ljBSPJeENOw2efoSWs416pr8/g9GOmZqUVFQMfr/z//r+dXk7h9vxeDCLuZqsF4CNDOzGv0wU75zHhlgE3II+mBznc+g6xGnMwA8n4l" } }, { "type": "step", "result": "=-\\frac{e^{u}u^{2}}{8^{3}}" }, { "type": "step", "primary": "$$8^{3}=512$$", "result": "=-\\frac{e^{u}u^{2}}{512}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{e^{u}u^{2}}{512}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s711kPpxvREthXvSVavuNs6mk3hxk9aCfAWodBRxXgUex8EuHmP+Mlx3UWu/2z/XFLDSlYx+wlzV79nT5r/AVmqJeGdZmS9Y7caQDLTm+4O7Tq/pVDk+uqUp4Mh+8MmSqwPUZDHRRVkrDOQpTZTNW6opSBv6izheLVUKQ/emokAUyEKC4SmDfgdaB0h/PWlULm7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=-\\int\\:-\\frac{e^{u}u^{2}}{512}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\left(-\\frac{1}{512}\\cdot\\:\\int\\:e^{u}u^{2}du\\right)" }, { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:e^{u}u^{2}du", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=u^{2}$$" }, { "type": "step", "primary": "$$v'=e^{u}$$" }, { "type": "interim", "title": "$$u'=\\left(u^{2}\\right)^{\\prime}=2u$$", "input": "\\left(u^{2}\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7K+rBgZSrXjr2hGKHMq+UqrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqoBkfwVA6f79TZE9da+mn3/tjL5hxvw/Kbqo5WiGHyvWytXHiku7Te0t0g+4a8q2Xw" } }, { "type": "interim", "title": "$$v=\\int\\:e^{u}du=e^{u}$$", "input": "\\int\\:e^{u}du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{u}du=e^{u}$$", "result": "=e^{u}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=e^{u}+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s764Ku1cMAzfBMQZ5NWuosU557+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuWEH8zsBqPdtVUPbEkgYIb65PR8UBYYTQDN+3rKN9QZo3Psiw7guEZF+7XAMh0QgNA==" } }, { "type": "step", "result": "=u^{2}e^{u}-\\int\\:2ue^{u}du" } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/gTxAYa8/WcSMXaESb9XwArnvpYQ0aygJfymuicdqpZRxIxZfG55Za+zoTmUSVss6YslwX7z6t6I+Le05fWU2pcignBBZFhWvjF+ywtUN2RHDLwXi6La7+0FEGTu6WJ8nZ04JX0uaeiV4M8qp8wkzaTG/7wDB5Csv/eG5mrahB0eSQilmHh9Rr5YjxdBgITC6VT/7GP0NzWpRUUaoY+Ghg=" } }, { "type": "step", "result": "=-\\left(-\\frac{1}{512}\\left(u^{2}e^{u}-\\int\\:2ue^{u}du\\right)\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\left(-\\frac{1}{512}\\left(u^{2}e^{u}-2\\cdot\\:\\int\\:ue^{u}du\\right)\\right)" }, { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:ue^{u}du", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=u$$" }, { "type": "step", "primary": "$$v'=e^{u}$$" }, { "type": "interim", "title": "$$u'=u^{\\prime}=1$$", "input": "u^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$u^{\\prime}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79RKcsDqucW74TALXtzJI7KboRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wrOr42DoRa4lmvtW+XTBjyKcmjY98EUG1lu/OC9CXbviq/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$v=\\int\\:e^{u}du=e^{u}$$", "input": "\\int\\:e^{u}du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{u}du=e^{u}$$", "result": "=e^{u}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=e^{u}+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s764Ku1cMAzfBMQZ5NWuosU557+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuWEH8zsBqPdtVUPbEkgYIb65PR8UBYYTQDN+3rKN9QZo3Psiw7guEZF+7XAMh0QgNA==" } }, { "type": "step", "result": "=ue^{u}-\\int\\:1\\cdot\\:e^{u}du" }, { "type": "interim", "title": "Simplify", "input": "ue^{u}-\\int\\:1\\cdot\\:e^{u}du", "result": "=e^{u}u-\\int\\:e^{u}du", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{u}=e^{u}$$", "result": "=e^{u}u-\\int\\:e^{u}du" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zRsNVBF7GMx1EIjug18FnqFjYEyrvCWvj0UgozEWLeQeSQilmHh9Rr5YjxdBgITC8051+98xgjCy9hiyp0OchubNA1dxl2kb16ubxL5hQyKYJ8UNzVgeR040e4kq2cn7KqQhfqSx4Ip0S2H5wgufDI0B4tKj1rjo3lSvhuHyFWXtgUxf6TtJKhZ0O9mS/RSDw==" } }, { "type": "step", "result": "=-\\left(-\\frac{1}{512}\\left(u^{2}e^{u}-2\\left(e^{u}u-\\int\\:e^{u}du\\right)\\right)\\right)" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{u}du=e^{u}$$", "result": "=-\\left(-\\frac{1}{512}\\left(u^{2}e^{u}-2\\left(e^{u}u-e^{u}\\right)\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=-8x$$", "result": "=-\\left(-\\frac{1}{512}\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(e^{-8x}\\left(-8x\\right)-e^{-8x}\\right)\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-\\left(-\\frac{1}{512}\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(e^{-8x}\\left(-8x\\right)-e^{-8x}\\right)\\right)\\right):{\\quad}\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)$$", "input": "-\\left(-\\frac{1}{512}\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(e^{-8x}\\left(-8x\\right)-e^{-8x}\\right)\\right)\\right)", "result": "=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1}{512}\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)\\right)}{512}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)\\right)=e^{-8x}\\left(8x\\right)^{2}-2\\left(-8e^{-8x}x-e^{-8x}\\right)$$", "input": "1\\cdot\\:\\left(\\left(-8x\\right)^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\left(-8x\\right)^{2}=\\left(8x\\right)^{2}$$", "input": "\\left(-8x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-8x\\right)^{2}=\\left(8x\\right)^{2}$$" ], "result": "=\\left(8x\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q48yg+G3VO8eY60AeMV2x913jtrSFDx+UNsawjlOjV1wN5JagZ/8xlmhLmWQGh6bP8vQyhiD4JSfqjIvcQ7tijSg806XI7vv3DjoG9mTAWqPATzsdMJbIBTfKlGugMwN" } }, { "type": "step", "result": "=1\\cdot\\:\\left(e^{-8x}\\left(8x\\right)^{2}-2\\left(-8e^{-8x}x-e^{-8x}\\right)\\right)" }, { "type": "step", "primary": "Refine", "result": "=\\left(8x\\right)^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73qwfwn6M3jv6zKzxEVgRV7aooSGMi5s8yy7hjFJyuPk2ka6SYHGlS5NQWAQm5i3e/ui0V7GtK+tTgPr391Bv2QOfOVs9mPIqDLV5QIWwt3n0pAWelvJELg2phQOv80NrOj/BpocfibpiIvA7OzQXkSNBw8nf7YXOv0VoUo+XxaPvls2yTaDWUcgETPUU6pidTEDxvTOJR9Tx0Irke3O2C7aooSGMi5s8yy7hjFJyuPk2ka6SYHGlS5NQWAQm5i3eyN3mElOrhjzM9FAs5OijGFrE4ChnWs0A4k2iMatu6cMVdQMWyWTnA7+CdGwiWw1mvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "=\\frac{e^{-8x}\\left(8x\\right)^{2}-2\\left(-8e^{-8x}x-e^{-8x}\\right)}{512}" }, { "type": "interim", "title": "$$\\left(8x\\right)^{2}e^{-8x}=64e^{-8x}x^{2}$$", "input": "\\left(8x\\right)^{2}e^{-8x}", "steps": [ { "type": "interim", "title": "$$\\left(8x\\right)^{2}=8^{2}x^{2}$$", "input": "\\left(8x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=8^{2}x^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hr+mSGTWh7fF3dyICh8ges0ag8T1MwTer44+aCS/ZFCeqWU0ybXg1YcOK7DwWF9m/z//r+dXk7h9vxeDCLuZqvyLzcXwo1XxefEQSKddxjnF3DenEWojLSGXYMDcAl7b" } }, { "type": "step", "result": "=8^{2}e^{-8x}x^{2}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=64e^{-8x}x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z1Nv+Hp8fxx9IM7Nyml5IHyRHuGw7+tM5METTDj6vVEZ9VrfiTRdKyyZL2pbraaWDCINaSiL/dovli8BGeXjhBGINW0cj7zNl67lBrvcmf0Q3rkV4+JiJGXhzqfOaq1MKerZXcY4iUujr3TKXY9/eQ==" } }, { "type": "step", "result": "=\\frac{64e^{-8x}x^{2}-2\\left(-8e^{-8x}x-e^{-8x}\\right)}{512}" }, { "type": "interim", "title": "Factor $$64x^{2}e^{-8x}-2\\left(-e^{-8x}8x-e^{-8x}\\right):{\\quad}2e^{-8x}\\left(32x^{2}+1+8x\\right)$$", "input": "64x^{2}e^{-8x}-2\\left(-e^{-8x}\\cdot\\:8x-e^{-8x}\\right)", "result": "=\\frac{2e^{-8x}\\left(32x^{2}+1+8x\\right)}{512}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:32x^{2}e^{-8x}-2\\left(-e^{-8x}-e^{-8x}\\cdot\\:8x\\right)" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(32x^{2}e^{-8x}-\\left(-e^{-8x}-e^{-8x}\\cdot\\:8x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$32e^{-8x}x^{2}-\\left(-8e^{-8x}x-e^{-8x}\\right):{\\quad}e^{-8x}\\left(32x^{2}+1+8x\\right)$$", "input": "32x^{2}e^{-8x}-\\left(-e^{-8x}-e^{-8x}\\cdot\\:8x\\right)", "result": "=2e^{-8x}\\left(32x^{2}+8x+1\\right)", "steps": [ { "type": "interim", "title": "Factor $$-e^{-8x}-e^{-8x}8x:{\\quad}-e^{-8x}\\left(1+8x\\right)$$", "input": "-e^{-8x}-e^{-8x}\\cdot\\:8x", "steps": [ { "type": "step", "primary": "Factor out common term $$e^{-8x}$$", "result": "=-e^{-8x}\\left(1+8x\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=32e^{-8x}x^{2}+e^{-8x}\\left(8x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$e^{-8x}$$", "result": "=e^{-8x}\\left(32x^{2}+1+8x\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{e^{-8x}\\left(32x^{2}+8x+1\\right)}{256}" }, { "type": "step", "result": "=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AIFT2vnAg4xm9hOxc49ZIQDCcYaIeAGY/oaslNj3WFC0sZSk8z2vV2z3ZK65DhAKnT42+1oRlGDXZTGCNzq8BVXTSum/z5kLpMzXS1UJIezTENL5jlqPSGX4Xou8NoCW+mxXk94dYwlzhQcf3hMuUryw/R6ZacICrhW8/5QipLKjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIERBEcLs16u4pNMjSsmDddd5mCFsPR1S4wroRMnLiDjV5eQphTXKZTbojLDSut81wfyGBnrMNyajNNThF1AXxU4=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "e^{-8x}ν=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)+c_{1}" }, { "type": "interim", "title": "Isolate $$ν:{\\quad}ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}$$", "input": "e^{-8x}ν=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)+c_{1}", "steps": [ { "type": "interim", "title": "Divide both sides by $$e^{-8x}$$", "input": "e^{-8x}ν=\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)+c_{1}", "result": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}", "steps": [ { "type": "step", "primary": "Divide both sides by $$e^{-8x}$$", "result": "\\frac{e^{-8x}ν}{e^{-8x}}=\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}+\\frac{c_{1}}{e^{-8x}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{e^{-8x}ν}{e^{-8x}}=\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}+\\frac{c_{1}}{e^{-8x}}", "result": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{e^{-8x}ν}{e^{-8x}}:{\\quad}ν$$", "input": "\\frac{e^{-8x}ν}{e^{-8x}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$e^{-8x}$$", "result": "=ν" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s745fZR6oHk9yRi4W6wJMvbHjpIy9GalwlVrTxd77WdNwgJ/ZZA32ZInFBpDtxBfiKAi+o8FY2dMrV7WTYXaeHyoEFMST8lDZxn1Yq5HMKVTtAb51BceE+a5ppiVMbLOTZnyT8Y1l2BaWniOZQq9UhrF50g8r2hEJi55PtRc69hDQ=" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}+\\frac{c_{1}}{e^{-8x}}:{\\quad}\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}$$", "input": "\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}+\\frac{c_{1}}{e^{-8x}}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}=\\frac{32x^{2}+8x+1}{256}$$", "input": "\\frac{\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)}{e^{-8x}}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)\\::{\\quad}\\frac{e^{-8x}\\left(32x^{2}+8x+1\\right)}{256}$$", "input": "\\frac{1}{256}e^{-8x}\\left(32x^{2}+8x+1\\right)", "result": "=\\frac{\\frac{e^{-8x}\\left(32x^{2}+8x+1\\right)}{256}}{e^{-8x}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{-8x}\\left(32x^{2}+8x+1\\right)}{256}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{-8x}=e^{-8x}$$", "result": "=\\frac{e^{-8x}\\left(32x^{2}+8x+1\\right)}{256}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{e^{-8x}\\left(32x^{2}+8x+1\\right)}{256e^{-8x}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{-8x}$$", "result": "=\\frac{32x^{2}+8x+1}{256}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajUOSKRTdXkPOzHoQw2oUq6iWnJmoIEgsGGOsRw5xcda01cIBrForrhW6S8+oY11gc6uO77lnnV79llIPeBEfCMFweweTrjGgjHNekryo/q6+R+5PqiRjy9vdxLY+eFenGQeTV4u20zW11V5PLOwL6WuEeGnrOKPdXwB981NMoEu4xFu1q8rdMGPStWY/qkyAw+qr7G32L2OtdgowtVVFpGBOSAS0lQhJCk32dTboGkZfkgqPhtyr0JU8nhS/1J8v9g==" } }, { "type": "interim", "title": "$$\\frac{c_{1}}{e^{-8x}}=c_{1}e^{8x}$$", "input": "\\frac{c_{1}}{e^{-8x}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$e^{-8x}=\\frac{1}{e^{8x}}$$" ], "result": "=\\frac{c_{1}}{\\frac{1}{e^{8x}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{c_{1}e^{8x}}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=c_{1}e^{8x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFapZl6h6k27790x8zBueJDnHktVoNtLjU9nOkmUWf91g99dC9fj9sg0EHzBIRDROUq7mmMbt75zw4jTax80Bz/L0MoYg+CUn6oyL3EO7Yp2J3RODthmtyxSwz1tPA6MjLh3miv7txY2iTgtT9M0R2vbp22yN0+LgmMk2HAaaus=" } }, { "type": "step", "result": "=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajUOSKRTdXkPOzHoQw2oUq6iWnJmoIEgsGGOsRw5xcda0hT19tsphUzzuNkhLydm2SoFR9agTnOBjYDRgYmCX4ebehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgppVVmOvmR+/adiegtvnU28cIiRkKkce3gUy3g+2PmYKTZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz1dVbNanNsboBJ234rlUVQEQ5IpFN1eQ87MehDDahSrqJacmaggSCwYY6xHDnFx1rSFPX22ymFTPO42SEvJ2bZKBvu040CR1+1tYL/6I6y4Zw==" } }, { "type": "step", "result": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Substitute back $$ν=y^{-1}:{\\quad}y^{-1}=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}$$", "input": "ν=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}", "steps": [ { "type": "step", "primary": "Substitute back $$ν=y^{-1}$$", "result": "y^{-1}=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" } ], "meta": { "interimType": "Generic Substitute Back Specific 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}$$", "input": "y^{-1}=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}", "steps": [ { "type": "step", "primary": "Refine", "result": "\\frac{1}{y}=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}" }, { "type": "interim", "title": "Multiply both sides by $$y$$", "input": "\\frac{1}{y}=\\frac{32x^{2}+8x+1}{256}+c_{1}e^{8x}", "result": "1=\\frac{\\left(32x^{2}+8x+1\\right)y}{256}+c_{1}e^{8x}y", "steps": [ { "type": "step", "primary": "Multiply both sides by $$y$$", "result": "\\frac{1}{y}y=\\frac{32x^{2}+8x+1}{256}y+c_{1}e^{8x}y" }, { "type": "step", "primary": "Simplify", "result": "1=\\frac{\\left(32x^{2}+8x+1\\right)y}{256}+c_{1}e^{8x}y" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Switch sides", "result": "\\frac{\\left(32x^{2}+8x+1\\right)y}{256}+c_{1}e^{8x}y=1" }, { "type": "interim", "title": "Multiply both sides by $$256$$", "input": "\\frac{\\left(32x^{2}+8x+1\\right)y}{256}+c_{1}e^{8x}y=1", "result": "\\left(32x^{2}+8x+1\\right)y+256c_{1}e^{8x}y=256", "steps": [ { "type": "step", "primary": "Multiply both sides by $$256$$", "result": "\\frac{\\left(32x^{2}+8x+1\\right)y}{256}\\cdot\\:256+c_{1}e^{8x}y\\cdot\\:256=1\\cdot\\:256" }, { "type": "step", "primary": "Simplify", "result": "\\left(32x^{2}+8x+1\\right)y+256c_{1}e^{8x}y=256" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Expand $$y\\left(32x^{2}+8x+1\\right):{\\quad}32x^{2}y+8xy+y$$", "input": "y\\left(32x^{2}+8x+1\\right)", "result": "32x^{2}y+8xy+y+256c_{1}e^{8x}y=256", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=y\\cdot\\:32x^{2}+y\\cdot\\:8x+y\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "result": "=32x^{2}y+8xy+1\\cdot\\:y" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:y=y$$", "result": "=32x^{2}y+8xy+y" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KNd7znz3MO57W4kKVD1r79MvHyY50dhXPFfrjcmooUhw1zW7EQZA5BZi2tDciCN2v+mUm3yaPvvy7kZWZKDeKO9sGZu5A1MXROmEpnxG69r4oEGVIGSzXY2KVbsX/ZUvpIa2OrHRvsYt8xjUyslMoQ==" } }, { "type": "interim", "title": "Factor $$32x^{2}y+8xy+y+256c_{1}e^{8x}y:{\\quad}y\\left(32x^{2}+8x+1+256c_{1}e^{8x}\\right)$$", "input": "32x^{2}y+8xy+y+256c_{1}e^{8x}y", "steps": [ { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(32x^{2}+8x+1+256c_{1}e^{8x}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "y\\left(32x^{2}+8x+1+256c_{1}e^{8x}\\right)=256" }, { "type": "interim", "title": "Divide both sides by $$32x^{2}+8x+1+256c_{1}e^{8x}$$", "input": "y\\left(32x^{2}+8x+1+256c_{1}e^{8x}\\right)=256", "result": "y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}", "steps": [ { "type": "step", "primary": "Divide both sides by $$32x^{2}+8x+1+256c_{1}e^{8x}$$", "result": "\\frac{y\\left(32x^{2}+8x+1+256c_{1}e^{8x}\\right)}{32x^{2}+8x+1+256c_{1}e^{8x}}=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}" } ], "meta": { "interimType": "ODE Solve Bernoulli 0Eq" } }, { "type": "step", "result": "y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=\\frac{256}{32x^{2}+8x+1+256c_{1}e^{8x}}" } } }, "meta": { "showVerify": true } }