Question

Solve the following differential equation:

(x² – y²)dx + 2xy dy = 0

Answer 1

Differential equation is

(x² – y²) dx + 2xy dy = 0

`\implies dy/dx = (-(x^2 - y^2))/(2xy)`   ...(1)

Which is a Homogeneous differential equation then put y = Vx in (1)

By (1) `d/dx (Vx) = ((Vx)^2 - x^2)/(2x(Vx))`

`\implies  V + x (dV)/dx = (V^2 - 1)/(2V)`

`\implies x  (dV)/dx = (V^2 - 1)/(2V) - V`

`\implies x (dV)/dx = (V^2 - 1 - 2V^2)/(2V)`

`\implies x (dV)/dx = (-V^2 - 1)/(2V)`

`\implies (2V)/(V^2 + 1) dV = - dx/x`

Now integrating both sides

`int (2V)/(V^2 + 1)dV = -int dx/x`

`\implies` log (V² + 1) = – log x + log c

`\implies log(V^2 + 1) = log(C/x)`

`\implies` V² + 1 = `C/x`

`\implies y^2/x^2 + 1 = C/x`

`\implies` y² + x² = Cx.