Chapter 2, Problem 5E

Using the Galerkin method, solve the following differential equation with the approximate solution in the form of $\tilde{u}\left(x\right)=c_{1}x+c_2{x}^2$. Compare the approximate solution with the exact one by plotting them on a graph. Also, compare the derivatives $\frac{du}{dx}$ and $\frac{d\tilde{u}}{dx}$.

$\begin{array}{l}\frac{d^{2}u}{d{x}^2}+x^2=0,0\le x\le 1\\ \begin{array}{c}u\left(0\right)=0\\ \frac{du}{dx}\left(1\right)=1\end{array}\}\text{ Boundary conditions}\end{array}$