Q$2$: For the following boundary$-$value problem:

$[(1+2x)w^{\text{'}\text{'}}{]}^{\text{'}\text{'}}+(x{w}^{\text{'}}{)}^{\text{'}}-x^2=0,w(0)=w^{\text{'}\text{'}}(0)=w(1)=w^{\text{'}\text{'}}(1)=0$

Use the Rayleigh$-$Ritz method and assume $w(x)={\displaystyle \underset{i=1}{\overset{N}{}}}{c}_i{}_i(x)$ with ${}_i(x)=x^{i+1}$ and $N=3$ to solve for $w(x).$ is a constant.