Chapter 2, Problem 15E

The boundary-value problem for a cantilevered beam can be written as

$\begin{array}{l}\frac{d^{4}w}{d{x}^4}-p\left(x\right)=0,0\le x\le 1\\ w\left(0\right)=\frac{dw}{dx}\left(0\right)=0,\frac{d^{2}w}{d{x}^2}\left(1\right)=1,\frac{d^{3}w}{d{x}^3}\left(1\right)=-1\end{array}$

Assume $p\left(x\right)=x$. Assuming the approximate deflection in the form of $\tilde{w}\left(x\right)=c_1{\varphi }_1\left(x\right)+c_2{\varphi }_2\left(x\right)=c_1{x}^2+c_2{x}^3$. Solve for the boundary-value problem using the Galerkin method. Compare the approximate solution to the exact solution by plotting the solutions on a graph.