How to solve de differential equation $u'' + (1+x²)u=-1$? How to solve the differential equation:
$$\frac{\mathrm{d}^2 u}{\mathrm{d} x^{2}} + (1+x^{2})u = -1$$ 
with $\frac{\mathrm{d}u}{\mathrm{d} x}(0) = 0$ and $u(1) = 0$.
I tried Laplace and Fourier Transforms but they simple end up in new diff equations that are nonlinear and very similar to the original. 
Trying using Taylor series I could not find an answer too.
The best I could get was looking for a power series solution:
$u(x) = \sum_{n=0}^{\infty} c_{n}x^{n}$.
However I still can't get an exact formula for the solution. 

This differential equation has a solution in terms of the Parabolic cylinder functions. You may solve for $\nu$ first, upon a suitable change of variables. 
Once you solve for the homogenous part, say $y_h = A y_1(x) + B y_2(x)$, you may find a particular solution trying solutions of the form $y = A(x) y_1(x)$ (or $y = B(x)y_2(x)$). I'm afraid that the resulting integrals are going to be a nightmare.
You can also find an approximation to $y(x)$ using WKB theory (have a look to Bender & Orszag).
Cheers!
PS: You may find useful some of the answers to this question.
Here's a numerical solution to your problem using Matlab:

