Chapter 2, Problem 13E

Using the Galerkin method, calculate the approximate solution of the following differential equation in the form of $u\left(x\right)=c_{1}x+c_2\frac{1}{2}{x}^2$.

$\begin{array}{l}\frac{d}{dx}\left(x\frac{du}{dx}\right)=x0\le x\le 1\\ u\left(0\right)=0\\ \frac{du}{dx}\left(1\right)=1\end{array}$