Separation of variables / integrating factor problem

[scarlets99]

Homework Statement

Find the general solution for this differential equation.

dy −2x^2 + y^2 + x
dx = x y

Homework Equations

The Attempt at a Solution

dy/dx = (−2x^2 + y^2 + x) / (x y)

let y^2 = v

dy/dx = v + x dv/dx

v + x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / v x^2

=> x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / (v x^2) - (v)

=> x (dv/dx) = [ (-2x^2 + v^2 x^2 + x ) - v^2 x^2 ] / vx^2

=> x (dv/dx) = [ -2x^2 + x ] / vx^2

=> dv/dx = (-2x + 1) / vx^2

separating variables

v dv = [ (-2x + 1) / x^2 ] dx

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

integrating

(1/2)v^2 = - 2 ln x - (1/x) + c

v^2 = -4 ln x - (2/x) + C

substitute v = y/x

y^2 / x^2 = -4ln x - (2/x) + C

y^2 = -4x^2 ln(x) - 2x + Cx^2

 

[anathema]

y = ± √(-4x^2 ln(x) - 2x + Cx^2)

This is the general solution for the given differential equation.