If you haven't already, finish the problem we started in class on the Simple Harmonic Oscillator. In particular, modify your $2$nd$-$order and $4$th$-$order Runge$-$Kutta functions to accept a system of equations $($in matrix form, similar to the example program on Canvas$).$ Then use both functions to solve the equation of motion for the simple harmonic oscillator:

$\frac{d^{2}x}{d{t}^2}=-{}^{2}x$

$($a$)$ The exact solution of the position of the simple harmonic oscillator is

$x=$Acos$(t)+$Bsin$(t).$

Solve for the coefficients A and B by hand, assuming $)=(0s$ and $)=(0s.$

$($b$)$ Make a single figure $($with four sub$-$panels$)$ showing the exact and numerical solutions for the following conditions using both your $2$nd$-$order and $4$th$-$order Runge$-$Kutta functions. $($Each plot should have three lines.$)$

i$.=0\mathrm{.}1,v_0=1\mathrm{.}0,x_0=0,h=1$

ii$.=0\mathrm{.}1,v_0=1\mathrm{.}0,x_0=0,h=5$

iii. $=0\mathrm{.}1,v_0=1\mathrm{.}0,x_0=0,h=10$

iv$.=0\mathrm{.}1,v_0=1\mathrm{.}0,x_0=0,h=50$