A popular ODE integration method is the Runge$-$Kutta $44$ method where, when applied to the ODE: $=(,),$ can be expressed as follows:

$+1=+\backslash$Delta $(1+22+23+4)6$

where the time step size is denoted by $\backslash$Delta $$ and with $1=(,)$

$2=(+\backslash$Delta $,+\backslash$Delta $1)22$

$3=(+\backslash$Delta $,+\backslash$Delta $2)22$

$3=(+\backslash$Delta $,+\backslash$Delta $3)$

In what follows, you will apply that method to the following ODE: $=$ with initial value

$(0)=1$ and with $>0.$ That is$,$ the generic function is simply: $(,)=.$

$1.$ Compute the gain factor

$()=+1$

as a function of $=\backslash$Delta $$ and plot it for $$ in $[0,4].$

$2.$ Determine the range of time step sizes $\backslash$Delta $$ for which the method is stable $($you may have

to use a root finding method for this$).$

$3.$ Is the method explicit or implicit? Justify. Recall that an explicit method calculates the

system status at a future time from the currently known system status. An implicit method calculates the system status at a future time from the system statuses at present and future times. If every step is explicit, then the method as a whole is explicit.

$4.$ Determine the range of time step sizes $\backslash$Delta $$ for which the method does not present spurious oscillations.

$5.$ Pick $=3$ sec$-1$ and write an ODE solver using the Runge$-$Kutta $44$ method. Pick the simulation duration time to be $20$ seconds. Plot the error, at time $20$ s$,$ between the exact solution and several numerical solutions, as a function of the time step sizes used. The numerical solutions should be obtained using $40,80,160,320,640,1280,2560,5120,$ and $10240$ time steps. The plot should be a log$-$log plot. Determine the convergence rate of the method $($i$.$e$.,$ the slope of that curve$).$