Consider the initial value problem

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

with the initial contition

$x(0)=1$

$($a$)$ Solve the equation, analytically.

$($b$)$ Use the Taylor$-$series method to solve the equation, given the stepsize $h=0\mathrm{.}10000.$ Provide two terms. Limit your results to five steps, with five significant figures.

$($c$)$ Use the simple Euler method to solve the equation, given the stepsize $h=0\mathrm{.}10000.$ Limit your results

$(5)$ to five steps, with five significant figures.

$($d$)$ 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.

$($e$)$ 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.

$[25]$