General Solution of the ODE $\eta' - 2 \zeta \eta = K\left(\zeta^2+1\right)^2$ 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.

Changing notations, let us consider the differential equation $$y'-2x y=K(x^2+1)^2$$ Using the solution of the homogeneous equation, just as Willie Wong suggested,  let $y=z\ e^{x^2}$ which reduces the equation to $$ e^{x^2}\, z'=K(x^2+1)^2$$ So, we are left with the problem of computing integrals $$J_n=\int x^{2n}\,e^{-x^2}\,dx$$ $(n=0,1,2).$ We find, using integration by parts, $$J_0=\frac{\sqrt{\pi }}{2} \, \text{erf}(x)$$ $$J_1=\frac{\sqrt{\pi }}{4} \, \text{erf}(x)-\frac{1}{2} e^{-x^2} x$$ $$J_2=\frac{3\sqrt{\pi }}{8} \, \text{erf}(x)-e^{-x^2} \left(\frac{x^3}{2}+\frac{3
   x}{4}\right)$$ Combining all the above $$z=K \left(\frac{11\sqrt{\pi }}{8}  \,\text{erf}(x)-e^{-x^2}
   \left(\frac{x^3}{2}+\frac{7 x}{4}\right)\right)+C$$ then $y$.
For the next step $\int y\,dx$, it seems to be much more complex and I suppose that part of the result would include some hypergeometric functions.
