How to find the general solution for $x y (\frac{d y}{d x} - 1) = x^{2} + y^{2}$ $xy (dy/dx - 1)= x^2 + y^2$ ?

Question

Homogeneous differential equations

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Answer 1 · 2018-05-20T12:16:01

$\frac{y}{x} = ln | y + x | + C$ $y/x=ln|y+x|+C$

Given

$xy(y'-1)=x^2+y^2$

Rearrange:

$y'=1+x/y+y/x$

Apply the substitution $y(x)=xv(x)$ :

$v+xv'=1+1/v+v$

Simplify and collect like terms:

$v/(v+1)v'=1/x$

Form the integral:

$int(1-1/(v+1))dv=int1/xdx$

Integrate term by term:

$v-ln|v+1|=ln|x|+C$

Simplify:

$v=ln|xv+x|+C$

Reverse the substitution:

$y/x=ln|y+x|+C$