Consider the initial value problem

$y^{\text{'}\text{'}}+x^{2}y=(x^2-4)sin(2x),x>0,$

$y(0)=0,y^{\text{'}}(0)=2.$

Discretize the above problem by the TR$-$BDF$2$ method. State your numerical scheme and initial conditions clearly.

Implement your scheme in Part $($c$)$ into Matlab code. For time step $k=\frac{1}{8},\frac{1}{16},\frac{1}{32},\frac{1}{64},\frac{1}{128},\frac{1}{256},$ compute the numerical error at $x=1.$ What's the convergence rate of your method?