Question

Find the general solution of yuy-xux=2

Answer 1

$\frac{dx}{-x}=\frac{dy}{y}=\frac{du}{2}$

1. First equation

$\frac{dx}{-x}=\frac{dy}{y}\\ \int \frac{dx}{-x}=\int \frac{dy}{y}\\ ln|xy|=ln]C|\\ xy=C;$

2 second equation

$\frac{dy}{y}=\frac{du}{2}\\ \int \frac{dy}{y}=\int \frac{du}{2}\\ ln|y|+C=\frac{u}{2}\\ u-ln(y^2)=C$

So we have two integrals of the charactestic system:

$xy=C_1;$

$u-ln(y^2)=C_2$

Therefore general solution has form

$F\left( xy, u-ln(y^2) \right)=0$

or $u=ln(y^2)+H\left( x\cdot y \right)$

where H(t)- any differentable function