Question

Find the general solution of the differential equation:

(xy – x²) dy = y² dx

Answer 1

(xy − x²) dy = y² dx

Divide throughout by y² (assuming y ≠ 0) to simplify:

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

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

To simplify, let us introduce a substitution. Divide the entire equation by x² (assuming x ≠ 0):

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

`v = y/x` (so that y = vx and dy = v dx + x dv)

Substitute into the equation:

`(1/xv - 1/v) (v dx + xdv) = 1/x^2 dx`

`(v/x - 1/(vx) (v dx + xdv) = 1/x^2 dx)`

Simplify step by step:

Multiply and separate:

`v^2/x dx + v/x xdv - 1/(vx) vdx - 1/(vx) xdv = 1/x^2 dx`

`v^2/x dx - 1/x dx + v dv - 1/vdv = 1/x^2 dx`

Group terms involving v and x. The equation becomes:

`(v^2 - 1)/v dv = (1/x^2 - (v^2-z)/x) dx`