$(30$ points$)$ Consider the differential equation:

$u(-4)=0on$

$u(0)=2,u_{WRM}(x)={\displaystyle \underset{j=1}{\overset{n}{}}}{c}_j{}_j(x)+{}_0(x)c_1{c}_2{x}_c=-\frac{3}{4}-\frac{8}{4}{c}_j(3-x{)}^2\frac{d^{2}u}{d{x}^2}=1-x,$:$(-4$

subjected $to$ the boundary conditions:

$u(-4)=0on$

$u(0)=2,$

Consider $an$ two$-$term parameter polynomial approximation $of$ the form

$u_{WRM}(x)={\displaystyle \underset{j=1}{\overset{n}{}}}{c}_j{}_j(x)+{}_0(x)$

$a)$ Determine the coefficients $c_1$ and $c_2$ and corresponding solutions using

$a.$ the Petrov$-$Galerkin method

$b.$ the Galerkin method

$c.$ the least$-$squares method

$d.$ the collocation method with two points $at{x}_c=-\frac{3}{4}$ and $-\frac{8}{4}$

$b$