Question

Find the general solution of the following differential equation:

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

Answer 1

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

Divide by x²:

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

Put y = vx. Then,

`(dy)/(dx) = v + x(dv)/(dx)`

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

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

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

Integrate:

tan⁻¹v = log |x| + c

Substitute `v = y/x`

∴ `tan^-1 (y/x) = log |x| + C`