We need to solve the given first order partial differential equation :
$(y-xu)u_x$ + $(x+yu)u_y$ = $x^{2} + y^{2}$ .
I tried this :
$\frac{dx}{y-xu}$ = $\frac{dy}{x+yu}$ = $\frac{du}{x^{2} + y^{2}}$ ,
First characteristic equation : $\frac{x.dx - y.dy}{-u(x^{2} + y^{2})}$ = $\frac{du}{x^{2} + y^{2}}$ , which gives , $C_1$ = $u^{2}+x^{2} - y^{2}$ ,
I was a little skeptical about the second equation , so kindly tell if its correct or not :
$\frac{y.dx + x.dy}{x^{2} + y^{2}}$ = $\frac{du}{x^{2} + y^{2}}$ , which gives , $C_2$ = $u-2xy$ ,
Are these constants correct ? Please help !
