Show that `y^(2)dx + (xy + x^(2))dy = 0` is a homogeneous differential equation. Also find its general solution.

Explanation

`(dy)/(dx) = -(y^(2))/(xy + x^(2))`
Let `f(x,y) = -(y^(2))/(xy + x^(2))`
Now `f(lambda x, lambda y) = -(lambda^(2) y^(2))/(lambda x.lambda y + lambda^(2) x^(2))`
`= - lambda^(0)(y^(2))/(xy + x^(2))`
`= lambda^(0)f(x, y)`
`:.` The given differential equation is the homogeneous differential equation.
Putting y = vx
`rArr (dy)/(dx) = v + x(dv)/(dx)`, we get
`v + x(dv)/(dx) = - (v^(2)x^(2))/(x.vx + x^(2))`
`rArr x(dv)/(dx) = (-v^(2) - v^(2) - v)/(v+1)`
`rArr (v+1)/(v(1+2v)) dv = -(dx)/(x)`
Integrating both sides, we get
`int ((1)/(v)-(1)/(1+2v))dv = -int (dx)/(x)`
`rArr ln |v| - (1)/(2)ln|(1+2v)| = -ln|x| + lnc`
`rArr (|vx|)/(sqrt(1+2v)) = c`
`rArr xy^(2) = c^(2)(x+2y)`
or `xy^(2) = k(x+2y)`