Problem $2-4$th Order Runge$-$Kutta

The $4$th Order Runge$-$Kutta Method can easily be implemented after you have created a $2$nd order Runge$-$Kutta method like you've done above. Take your code and modify it so that it will perform a standard $4$th order Runge$-$Kutta method where each integration step is taken by computing the following quantities sequentially.

$k_1=f(x_i,y_i)$

$k_2=f(x_i+0\mathrm{.}5h,y_i+0\mathrm{.}5k_1{h}_h)$

$k_3=f(x_i+0\mathrm{.}5h,y_i+0\mathrm{.}5k_2{h}_i)$

$k_4=f(x_i+h,y_i+k_{3}h)$

$=\frac{k_1+2k_2+2k_3+k_4}{6}$

$y_{i+1}=y_i+h$