# Substitution for diff. eq.

#### Croc

Hi

Having a bit of trouble with what the substition would be to solve this diff. Any ideas? Thanks.

dy/dx = (xy)/(x^2 - y^2)

#### 11rdc11

Hi

Having a bit of trouble with what the substition would be to solve this diff. Any ideas? Thanks.

dy/dx = (xy)/(x^2 - y^2)
divide the right hand side by x^2

$$\displaystyle \frac{dy}{dx} = \displaystyle{\frac{\frac{xy}{x^2}}{\frac{x^2}{x^2} - \frac{y^2}{x^2}}}$$

$$\displaystyle \frac{dy}{dx} = \displaystyle{\frac{\frac{y}{x}}{1-\bigg(\frac{y}{x}\bigg)^2}}$$

now let

$$\displaystyle v = \frac{y}{x}$$

and

$$\displaystyle vx =y$$

so

$$\displaystyle v + \bigg(\frac{dv}{dx}\bigg)x = \frac{dy}{dx}$$

now sub everything in and solve.