I am trying to find the function $\Phi(\zeta)$ which is determined by the ordinary differential equation $$\Phi'' - 2 \zeta \Phi' = K\left(\zeta^2+1\right)^2$$ where $K$ is a constant from my model, i will like to retain.
I am interested in finding the general solution of this ode. I was thinking if we assume $\Phi' = \eta$ then our ode will become.
$$\eta' - 2 \zeta \eta = K\left(\zeta^2+1\right)^2$$ which if i am not mistaken is still a non-linear ode.
How can we proceed here ? I am also open to other methods of solving this ODE.