Question

Solve the following differential equation:

x²dy + (xy + y²) dx = 0

Answer 1

`dy/dx = − ((xy + y^2)/x^2)`

Let y = vx and and `dy/dx = v + x(dv)/(dx)` into the equation

`v + x(dv)/(dx) = − (x (vx) + (vx)^2)/x^2`

v + x `(dv)/dx = − (vx^2 + v^2x^2)/x^2`

v + x `(dv)/dx = −(v + v^2)`

Separation of Variables Subtract v from both sides to isolate the dv terms:

x`(dv)/dx = −v −v^2 − v`

x`(dv)/dx = −(v^2 + 2v)`