$(60$ min.$)$ Consider the following boundary value problem:

Time left $0$:$33$:$08$

$\frac{d^{2}u}{(d)t^2}+7*\frac{du}{(d)t}-\sqrt[\phantom{2}]{u+5}=t$

$\frac{du}{dt}{|}_t=0=1,u(t=1\mathrm{.}6)=7$

Determine u$(0\mathrm{.}8)$ employing a $2$nd order $($or$,$ higher order$)$ Runge$-$Kutta $($RK$2)$ algorithm.

The step size $(t)$ should be set to a number that limits the truncation error for $u(0\mathrm{.}8)$ at $1\%$ or less $($Hint: check the relative change in solution by

trying $\frac{t}{2},$ or $\frac{t}{4}$ etc.$).$

Shortly explain how you solved this question. Prepare a short report including computer program, and$/$or screen shots etc.

Use of Excel, $($and its Solver$),$ Matlab, or any similar computer program for faster calculations is recommended.

Alternatives are accepted, but a second$-$order Runge Kutta method is:

RK$2($midpoint$)$ method:

$y_{i+1}=y_i+k_2*x$

$k_1=f(x_i,y_i)$

$k_2=f(x_i+(\frac{1}{2})x,y_i+(\frac{1}{2})k_1*x)$

Answer:?

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