Solve the differential equation: `(2a x+x^2)\ (dy)/(dx)=a^2+2a x`

Explanation

It is given,

`(2 a x+x^{2}) \frac{d y}{d x}=a^{2}+2 a x`

`d y=\frac{a(a+2 x)}{x(x+2 a)} d x`

Split into P.F.s.

`d y=\frac{a}{2}[\frac{1}{x}+\frac{3}{x+2 a}] d x`

Integrating we get,

`int d y=\int \frac{a}{2}[\frac{1}{x}+\frac{3}{x+2 a}] d x `

`y+k=\frac{a}{2}[\log x+3 \log (x+2 a)]`