How to solve this equation for $z(x)$?

Question

$xdy = (y - \sqrt{x^2 +y^2}) dx$

$y(x) = z(x) x$

$dy = zdx+xdz$

$x(zdx+xdz)=(zx-\sqrt{x^2 +x^2 z^2}) dx$

$x(zdx+xdz) =x(z-\sqrt{1+z^2})dx$

$(zdx+xdz)=(z-\sqrt{1+z^2})dx$

Answer 1

Observe that on the RHS, when you multiply the dx term with the ones in the bracket, the term zdx appears, which is also present in the LHS. Therefore they both cancel out, and now you are left with a simpler equation. Just rearrange them afterward, and the solution becomes easy.

• dx/x = dz/√(1+ z^2)
• ln(x) = ln( z + √(1+ z^2)) 1/x = z + √( 1+ z^2)