Answer to Question #222288 in Differential Equations for dante

Question

Solve the equation dy/dx= -(x²+y²)/xy

Answer 1

Let us solve the equation $\frac{dy}{dx}= -\frac{x^2+y^2}{xy}$ using the transformation $y=ux.$ Then $\frac{dy}{dx}=\frac{du}{dx}x+u,$ and hence we get the equation $\frac{du}{dx}x+u= -\frac{1+u^2}{u},$ which is equivalent to $\frac{du}{dx}x= -\frac{1+u^2}{u}-u,$ and hence to $\frac{du}{dx}x= -\frac{1+2u^2}{u}.$

It follows that

$\frac{udu}{1+2u^2}= -\frac{dx}{x}$

$\int\frac{udu}{1+2u^2}= -\int\frac{dx}{x}$

$\frac{1}{4}\int\frac{4udu}{1+2u^2}= -\int\frac{dx}{x}$

$\int\frac{d(1+2u^2)}{1+2u^2}= -4\int\frac{dx}{x}$

$\ln(1+2u^2)= -4\ln|x|+\ln|C|$

$\ln(1+2u^2)+4\ln|x|=\ln|C|$

$\ln(x^4(1+2u^2))=\ln|C|$

$x^4(1+2u^2)=C$

$x^4(1+2(\frac{y}{x})^2)=C$

We conclude that the general solution of the equation $\frac{dy}{dx}= -\frac{x^2+y^2}{xy}$ is

$x^4+2x^2y^2=C$