(dy}{dx}-(\frac{1+x)/x)y=(y^2)/x

{ "query": { "display": "$$\\frac{dy}{dx}-\\left(\\frac{1+x}{x}\\right)y=\\frac{y^{2}}{x}$$", "symbolab_question": "ODE#\\frac{dy}{dx}-(\\frac{1+x}{x})y=\\frac{y^{2}}{x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstBernoulli", "default": "y=\\frac{xe^{x}}{-e^{x}+c_{1}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{dy}{dx}-\\left(\\frac{1+x}{x}\\right)y=\\frac{y^{2}}{x}:{\\quad}y=\\frac{xe^{x}}{-e^{x}+c_{1}}$$", "input": "\\frac{dy}{dx}-\\left(\\frac{1+x}{x}\\right)y=\\frac{y^{2}}{x}", "steps": [ { "type": "interim", "title": "Solve Bernoulli ODE:$${\\quad}y=\\frac{xe^{x}}{-e^{x}+c_{1}}$$", "input": "\\frac{dy}{dx}-\\left(\\frac{1+x}{x}\\right)y=\\frac{y^{2}}{x}", "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)-\\left(\\frac{1+x}{x}\\right)y=\\frac{y^{2}}{x}" }, { "type": "interim", "title": "Rewrite in the form of a first order Bernoulli ODE", "input": "y^{\\prime}\\left(x\\right)-\\frac{1+x}{x}y=\\frac{y^{2}}{x}", "result": "y^{\\prime}\\left(x\\right)-\\frac{1+x}{x}y=\\frac{1}{x}y^{2}", "steps": [ { "type": "step", "primary": "Standard form of a first order Bernoulli ODE", "secondary": [ "$$y'+p\\left(x\\right)y=q\\left(x\\right)y^n$$" ] }, { "type": "step", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{1+x}{x}y=\\frac{y^{2}}{x}" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{\\left(1+x\\right)y}{x}=\\frac{y^{2}}{x}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$p\\left(x\\right)=-\\frac{1+x}{x},\\:{\\quad}q\\left(x\\right)=\\frac{1}{x},\\:{\\quad}n=2$$" ], "result": "y^{^{\\prime}}\\left(x\\right)-\\frac{1+x}{x}y=\\frac{1}{x}y^{2}" } ], "meta": { "interimType": "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 $$\\frac{1}{1-n}v'+p\\left(x\\right)v=q\\left(x\\right):{\\quad}-ν^{\\prime}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}$$", "input": "y^{\\prime}\\left(x\\right)-\\frac{1+x}{x}y=\\frac{1}{x}y^{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$y^{2}$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}-\\frac{\\frac{1+x}{x}y}{y^{2}}=\\frac{\\frac{1}{x}y^{2}}{y^{2}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}-\\frac{1+x}{xy}=\\frac{1}{x}" }, { "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{1+x}{xy}=\\frac{1}{x}" }, { "type": "step", "primary": "Substitute $$y^{-1}$$ with $$ν$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{y^{2}}-\\frac{1+x}{x}ν=\\frac{1}{x}" }, { "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}}-\\frac{1+x}{x}ν=\\frac{1}{x}" }, { "type": "step", "primary": "Simplify", "result": "-ν^{^{\\prime}}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}" } ], "meta": { "interimType": "Transform To Bernoulli 1Eq" } }, { "type": "interim", "title": "Solve $$-ν^{\\prime}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}:{\\quad}ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}$$", "input": "-ν^{\\prime}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}", "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}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}", "result": "ν^{\\prime}+\\frac{1+x}{x}ν=-\\frac{1}{x}", "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}}-\\frac{\\left(1+x\\right)ν}{x}=\\frac{1}{x}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "-\\frac{ν^{^{\\prime}}}{-1}-\\frac{\\frac{\\left(1+x\\right)ν}{x}}{-1}=\\frac{\\frac{1}{x}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "ν^{^{\\prime}}+\\frac{\\left(1+x\\right)ν}{x}=-\\frac{1}{x}" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$p\\left(x\\right)=\\frac{1+x}{x},\\:{\\quad}q\\left(x\\right)=-\\frac{1}{x}$$" ], "result": "ν^{^{\\prime}}+\\frac{1+x}{x}ν=-\\frac{1}{x}" } ], "meta": { "interimType": "Canon First Order ODE 2Eq" } }, { "type": "interim", "title": "Find the integration factor:$${\\quad}μ\\left(x\\right)=e^{x}x$$", "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": "$$p\\left(x\\right)=\\frac{1+x}{x}$$", "result": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{^{\\prime}}=\\frac{1+x}{x}" }, { "type": "interim", "title": "Solve $$\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=\\frac{1+x}{x}:{\\quad}μ\\left(x\\right)=e^{x+c_{1}}x$$", "input": "\\left(\\ln\\left(μ\\left(x\\right)\\right)\\right)^{\\prime}=\\frac{1+x}{x}", "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\\:\\frac{1+x}{x}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{1+x}{x}dx=\\ln\\left(x\\right)+x+c_{1}$$", "input": "\\int\\:\\frac{1+x}{x}dx", "steps": [ { "type": "interim", "title": "Expand $$\\frac{1+x}{x}:{\\quad}\\frac{1}{x}+1$$", "input": "\\frac{1+x}{x}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{1+x}{x}=\\frac{1}{x}+\\frac{x}{x}$$" ], "result": "=\\frac{1}{x}+\\frac{x}{x}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "secondary": [ "$$\\frac{x}{x}=1$$" ], "result": "=\\frac{1}{x}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TdxkTKDGyLba8sa+CvgttACWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQm8heGa3QDRZxOgSWy/w30FIEFMST8lDZxn1Yq5HMKVTvDpCb0LomfCc/BDlzuuIuG/uxKSHJs9S1oqoSPiOtibg==" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{x}+1dx" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:\\frac{1}{x}dx+\\int\\:1dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{x}dx=\\ln\\left(x\\right)$$", "input": "\\int\\:\\frac{1}{x}dx", "steps": [ { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{x}dx=\\ln\\left(x\\right),\\:$$assuming a complex-valued logarithm", "result": "=\\ln\\left(x\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\ln\\left(x\\right)+x" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left(x\\right)+x+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)=\\ln\\left(x\\right)+x+c_{1}" }, { "type": "interim", "title": "Isolate $$μ\\left(x\\right):{\\quad}μ\\left(x\\right)=e^{x+c_{1}}x$$", "input": "\\ln\\left(μ\\left(x\\right)\\right)=\\ln\\left(x\\right)+x+c_{1}", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(μ\\left(x\\right)\\right)=\\ln\\left(x\\right)+x+c_{1}", "result": "μ\\left(x\\right)=e^{x+c_{1}}x", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(μ\\left(x\\right)\\right)=\\ln\\left(x\\right)+x+c_{1}\\quad\\:\\Rightarrow\\:\\quad\\:μ\\left(x\\right)=e^{\\ln\\left(x\\right)+x+c_{1}}$$" ], "result": "μ\\left(x\\right)=e^{\\ln\\left(x\\right)+x+c_{1}}" }, { "type": "interim", "title": "Expand $$e^{\\ln\\left(x\\right)+x+c_{1}}:{\\quad}e^{x+c_{1}}x$$", "input": "e^{\\ln\\left(x\\right)+x+c_{1}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{\\ln\\left(x\\right)}e^{x+c_{1}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(x\\right)}:{\\quad}x$$", "input": "e^{\\ln\\left(x\\right)}", "result": "=xe^{x+c_{1}}", "steps": [ { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "result": "=x", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=e^{x+c_{1}}x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n8v9yoTt8HuIa/j9rJCj7KG1APzraWojl2EXi99h53m+sC8ksde3EL3dW7N7EScc6upJ8tRrcqdzpqw4uWy22Tu/0m71n2FfPReH2RI2dnj8bYA0b6V2RSTOZ7Os9NODf7fNk8t2EyVolmg+zl/ueNQPMmtpIekbY9TH2qp6FCs=" } }, { "type": "step", "result": "μ\\left(x\\right)=e^{x+c_{1}}x" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xZ/kvw9iKo4JlZMns/8rfzBRhWgZAs8cBKCp9DTiMQwmSzwWcMgxhWR3+PQclu59hqX8q7xCWwsRcJY5APRK6siNG73ekXpR3ysC7LzMiybtWE9F3hNcR6kIptQh4DXu1TEDYR7afb1dd8Vc+66YqU3kCh3oevUunZ7/b0qFKBSpWXSwya7DkA7lVz1D+b63sIjaxJ4DvjTb2fbKjbvtlQ==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=e^{x+c_{1}}x" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "μ\\left(x\\right)=e^{x+c_{1}}x" }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$e^{x+c_{1}}x=e^{x}e^{c_{1}}x$$" ], "result": "μ\\left(x\\right)=e^{x}e^{c_{1}}x" }, { "type": "step", "primary": "The constant $$e^{c_{1}}\\:$$can be dropped (it will be absorbed into C)", "result": "μ\\left(x\\right)=e^{x}x" } ], "meta": { "interimType": "Integrating Factor Top 0Eq" } }, { "type": "interim", "title": "Put the equation in the form $$\\left(\\mu\\left(x\\right){\\cdot}y\\right)'=\\mu\\left(x\\right){\\cdot}q\\left(x\\right):{\\quad}\\left(e^{x}xν\\right)^{\\prime}=-e^{x}$$", "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}}+\\frac{1+x}{x}ν=-\\frac{1}{x}" }, { "type": "step", "primary": "Multiply both sides by the integrating factor, $$e^{x}x$$", "result": "ν^{^{\\prime}}e^{x}x+\\frac{1+x}{x}νe^{x}x=\\left(-\\frac{1}{x}\\right)e^{x}x" }, { "type": "step", "primary": "Simplify", "result": "e^{x}xν^{^{\\prime}}+e^{x}\\left(1+x\\right)ν=-e^{x}" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{x}x,\\:g=ν:{\\quad}e^{x}xν^{\\prime}+e^{x}\\left(1+x\\right)ν=\\left(e^{x}xν\\right)^{\\prime}$$" ], "result": "\\left(e^{x}xν\\right)^{^{\\prime}}=-e^{x}" } ], "meta": { "interimType": "Bring Linear To Derivative Form Left 0Eq" } }, { "type": "interim", "title": "Solve $$\\left(e^{x}xν\\right)^{\\prime}=-e^{x}:{\\quad}ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}$$", "input": "\\left(e^{x}xν\\right)^{\\prime}=-e^{x}", "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^{x}xν=\\int\\:-e^{x}dx" }, { "type": "interim", "title": "$$\\int\\:-e^{x}dx=-e^{x}+c_{1}$$", "input": "\\int\\:-e^{x}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^{x}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{x}dx=e^{x}$$", "result": "=-e^{x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-e^{x}+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^{x}xν=-e^{x}+c_{1}" }, { "type": "interim", "title": "Isolate $$ν:{\\quad}ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}$$", "input": "e^{x}xν=-e^{x}+c_{1}", "steps": [ { "type": "interim", "title": "Divide both sides by $$e^{x}x$$", "input": "e^{x}xν=-e^{x}+c_{1}", "result": "ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}", "steps": [ { "type": "step", "primary": "Divide both sides by $$e^{x}x$$", "result": "\\frac{e^{x}xν}{e^{x}x}=-\\frac{e^{x}}{e^{x}x}+\\frac{c_{1}}{e^{x}x}" }, { "type": "step", "primary": "Simplify", "result": "ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}" } ], "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{1}{x}+\\frac{c_{1}}{e^{x}x}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}" } ], "meta": { "solvingClass": "ODE", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Substitute back $$ν=y^{-1}:{\\quad}y^{-1}=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}$$", "input": "ν=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}", "steps": [ { "type": "step", "primary": "Substitute back $$ν=y^{-1}$$", "result": "y^{-1}=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}" } ], "meta": { "interimType": "Generic Substitute Back Specific 1Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=\\frac{xe^{x}}{-e^{x}+c_{1}}$$", "input": "y^{-1}=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}", "steps": [ { "type": "step", "primary": "Refine", "result": "\\frac{1}{y}=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}" }, { "type": "interim", "title": "Multiply both sides by $$y$$", "input": "\\frac{1}{y}=-\\frac{1}{x}+\\frac{c_{1}}{e^{x}x}", "result": "1=-\\frac{y}{x}+\\frac{c_{1}y}{e^{x}x}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$y$$", "result": "\\frac{1}{y}y=-\\frac{1}{x}y+\\frac{c_{1}}{e^{x}x}y" }, { "type": "step", "primary": "Simplify", "result": "1=-\\frac{y}{x}+\\frac{c_{1}y}{e^{x}x}" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Switch sides", "result": "-\\frac{y}{x}+\\frac{c_{1}y}{e^{x}x}=1" }, { "type": "interim", "title": "Multiply by LCM", "input": "-\\frac{y}{x}+\\frac{c_{1}y}{e^{x}x}=1", "result": "-y+\\frac{c_{1}y}{e^{x}}=x", "steps": [ { "type": "interim", "title": "Find Least Common Multiplier of $$x,\\:e^{x}x:{\\quad}x$$", "input": "x,\\:e^{x}x", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$x$$ or $$e^{x}x$$", "result": "=x" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top in Equation Title 1Eq" } }, { "type": "step", "primary": "Multiply by LCM=$$x$$", "result": "-\\frac{y}{x}x+\\frac{c_{1}y}{e^{x}x}x=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "-y+\\frac{c_{1}y}{e^{x}}=x" } ], "meta": { "interimType": "Equation LCM Multiply Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDwxGZRSng4mWYMukryk5PuaDu3gRXrf1vjhcS2mXrE5pV6aL0otHIrA7AWCy2faei0hVm2ydBgPbIj+0jx7xSkBtqpgW9DIELTpLnppFbAo8vW4jeL6yawhE3sJ4m72r5N5Aod6Hr1Lp2e/29KhSgUSbOHyvzxlfz1y7043uXYRLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Multiply both sides by $$e^{x}$$", "input": "-y+\\frac{c_{1}y}{e^{x}}=x", "result": "-e^{x}y+c_{1}y=xe^{x}", "steps": [ { "type": "step", "primary": "Multiply both sides by $$e^{x}$$", "result": "-ye^{x}+\\frac{c_{1}y}{e^{x}}e^{x}=xe^{x}" }, { "type": "step", "primary": "Simplify", "result": "-e^{x}y+c_{1}y=xe^{x}" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$-e^{x}y+c_{1}y:{\\quad}y\\left(-e^{x}+c_{1}\\right)$$", "input": "-e^{x}y+c_{1}y", "steps": [ { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(-e^{x}+c_{1}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "y\\left(-e^{x}+c_{1}\\right)=xe^{x}" }, { "type": "interim", "title": "Divide both sides by $$-e^{x}+c_{1}$$", "input": "y\\left(-e^{x}+c_{1}\\right)=xe^{x}", "result": "y=\\frac{xe^{x}}{-e^{x}+c_{1}}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-e^{x}+c_{1}$$", "result": "\\frac{y\\left(-e^{x}+c_{1}\\right)}{-e^{x}+c_{1}}=\\frac{xe^{x}}{-e^{x}+c_{1}}" }, { "type": "step", "primary": "Simplify", "result": "y=\\frac{xe^{x}}{-e^{x}+c_{1}}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"#>#ODE#>#y=\\frac{xe^{x}}{-e^{x}+c_{1}}" } } }, "meta": { "showVerify": true } }