Solving a 2nd order ODE: $\frac{d^2}{dx^2}y(x)=\left(C+(1+x^2)^{-1}\right)y(x)$. I would like to solve the following ode:
$$\frac{d^2}{dx^2}y(x)=\left(C+(1+x^2)^{-1}\right)y(x),\quad x\in\mathbb{R};$$
with boundary condition $y(0)=1$. $C$ is just some constant.
I am very stuck with this. Does anyone have any suggestions of how to proceed?

I am afraid that a closed form solution could not exist and that, provided a second  boundary condition, numerical method would be required.
Even if $C=0$ the solution is far away to be simple since given by
$$y=\, _2F_1\left(-\frac{\sqrt{5}+1}{4} ,\frac{\sqrt{5}-1}{4}
   ;\frac{1}{2};-x^2\right)+c_1\, x \,\, _2F_1\left(-\frac{\sqrt{5}-1}{4},\frac{\sqrt{5}+1}{4};\frac{3}{2};-x^2\right)$$ where appear hypergeometric functions.
