Answer to Question #113702 in Differential Equations for usha

Question

Solve z = (x – y)∅ (x² + y²)

Answer 1

$z=(x-y)\phi(x^2+y^2)$

Let's denote $\partial z/\partial x=p, \,\, \partial z/\partial y=q$.

Differentiate both sides with respect to $x$ then to $y$:

$p=(x-y) \phi'(x^2+y^2)2x+\phi(x^2+y^2)$,

$q=(x-y) \phi'(x^2+y^2)2y-\phi(x^2+y^2)$.

Hence, we have

$\frac{p-\phi(x^2+y^2)}{q+\phi(x^2+y^2)}=\frac{x}{y} \rightsquigarrow py-qx=\phi(x^2+y^2)(x+y)$