Question

What is a solution to the differential equation `dy/dx=y^2/x^2+y/x+1`?

Answer 1

`y = xtan (ln x + C)`

Explanation:

This is a first order linear homogeneous equation. NB here homogeneous has its own meaning. it means that the equation can be written in form `y' = f(x,y)` and that `f(kx, ky) = f(x,y)` for constant k.

so standard approach is to let `v = y/x`

so
`y = v * x`
`y' = v' x + v`

thus, plugging this into the original.....

`v' x + v = v^2 + v + 1`

`v' x = v^2 + 1`

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

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

standard integral: `tan^(-1) v = ln x + C`

`tan^(-1) (y/x) = ln x + C`

`(y/x) = tan (ln x + C)`

`y = xtan (ln x + C)`