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  • splve x^{2}\frac{dy}{dx}=x^{2}+xy+y^{2}

solve x^{2}\frac{dy}{dx}=x^{2}+xy+y^{2}

Answers (1)

Given:

x^{2}\frac{dy}{dx}=x^{2}+y^{2}+xy

To find: solution for the differential equation

Rewriting the given equation as

\frac{dy}{dx}=1+\frac{y^{2}}{x^{2}}+\frac{y}{x}

Clearly, it is a homogenous equation

Assume y=vx

Differentiate on both sides

\frac{dy}{dx}=V+x\frac{dv}{dx}

Substituting dy/dx in the equation

1+\frac{y^{2}}{x^{2}}+\frac{y}{x}=V+x\frac{dV}{dx}\\ \\1+V^{2}+V=V+x\frac{dV}{dx}\\ \\1+V^{2}=x\frac{dV}{dx}\\\\ \frac{dx}{x}=\frac{dV}{1+V^{2}}

Integrating on both sides

\int \frac{dx}{x}=\int \frac{dV}{1+V^{2}}\\ \ln x=\arctan V+c\\\\ Formula: \int \frac{dx}{x}=\ln x+c\\\int \frac{dV}{1+V^{2}}=\tan^{-1}V+c\\ $Substitute $v=\frac{y}{x}\\ \ln x=\tan^{-1}\frac{y}{x}+c

 is the solution for the differential equation

 

Posted by

infoexpert24

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