Question $2$: $25$ Marks

Consider the initial value problem

$\frac{dx}{dt}=\frac{t}{x},0t5$

with the initial contition

$y(0)=1.$

$(2\mathrm{.}1)$ Solve the equation, analytically.

$(2\mathrm{.}2)$ Use the Taylor$-$series method $to$ solve the equation, given the stepsize $h=0\mathrm{.}10000.$

Limit your results $to$ five steps, with five significant figures.

$(2\mathrm{.}3)$ Use the simple Euler method $to$ solve the equation, given the stepsize $h=0\mathrm{.}10000.$

Limit your results $to$ five steps, with five significant figures.

$(2\mathrm{.}4)$ Use the modified Euler method $to$ solve the equation, given the stepsize $h=0\mathrm{.}10000.$

Limit your results $to$ five steps, with five significant figures.

$(2\mathrm{.}5)$ Use the second$-$oder Runge$-$Kutta $to$ solve the equation, given the stepsize $h=$

$0\mathrm{.}10000.$ Limit your results $to$ five steps, with five significant figures.