Solve udydy/udy+ydu=y+sqrt{u^2y^2-y^2}

$$u d y \frac { d y } { u d y + y d u } = y + \sqrt { u ^ { 2 } y ^ { 2 } - y ^ { 2 } }$$

Solve for d

$$d=\frac{2\left(|y|\sqrt{u^{2}-1}+y\right)}{y}$$

$$\left(y\neq 0\text{ and }u\neq -\sqrt{2}\text{ and }u\leq -1\right)\text{ or }\left(y\neq 0\text{ and }u\neq \sqrt{2}\text{ and }u\geq 1\right)\text{ or }\left(y>0\text{ and }|u|\geq 1\right)$$

Solve for u

$$u=\frac{\sqrt{d^{2}-4d+8}}{2}$$

$$u=-\frac{\sqrt{d^{2}-4d+8}}{2}\text{, }\left(d\neq 0\text{ and }d\leq 2\text{ and }y<0\right)\text{ or }\left(d\geq 2\text{ and }y>0\right)$$