$($a$)$ Given a function $f(x)$ create an m$-$function

function $[x,u]=$ finite$\_$difference $(\{$:$a,u_{-}a,b,u_{-}b,N)$

that solves the boundary value problem

$a,bN-1xu(x)uLU{u}^{\text{'}\text{'}}(x)+u(x)=cos(x)+xcos(2x)u(0)=0u(5)=1N=3,N=6,N=12N=24N{f}_1(x),f_2(x),f_3(x)f=f_{1}f=f_{2}f=f_3$cdots $in$ the vector $x$ and the approximation $of$ the solution $u(x)at$ these points

$in$ the vector $u.$ Hint: use the $LU$ factorization.

$(b)Inhw04q2.m$ use the code from $aa=x_1$

cdots $in$ the vector $x$ and the approximation $of$ the solution $u(x)at$ these points

$in$ the vector $u.$ Hint: use the $LU$ factorization.

$(b)Inhw04q2.m$ use the code from $a{u}^{\text{'}\text{'}}(x)+u(x)=f(x),a$

$on$ the interval $a,b$ cut into $N-1$ equal pieces and returns the points $a=x_1$

cdots $in$ the vector $x$ and the approximation $of$ the solution $u(x)at$ these points

$in$ the vector $u.$ Hint: use the $LU$ factorization.

$(b)Inhw04q2.m$ use the code from $a$