Task $2$: IVP Solvers $(15$ pts$)$

Write Python functions for the Euler, Midpoint, Heun, and Classic RK$-4$ methods of solving IVPs. Save all

functions to a file called A$4.$py$.$

Valid call: T$,$ Y $=$ euler$\_$integrate$($fun$,$ t$0,$ y$0,$ tStop, h$)$

Inputs: fun : $($callable$)$ ODE function handle

t$0$ : $($float$)$ Initial time step

y$0$ : $($float or $1$D numpy array$)$ Initial value of y

corresponding to t$0$

tStop : $($float$)$ Final time step

h : $($float$)$ Step size

Outputs: T : $(1$D numpy array$)$ integrated t$-$values

Y : $(1$D or $2$D numpy array$)$ integrated y$-$values

Assumptions: $$ The ODE function is called by: dydt $=$ fun$($t$,$y$)$

$$ h will be provided such that the solver will not step past

tStop

Note: Aside from the function names, the information above is the same for midpoint$\_$integrate,

heun$\_$integrate and RK$4\_$integrate.