Problem $2(25$ pts $).$ A sphere with radius $R$ has thermal energy $E=cT,$ where $T$ is its

absolute temperature $($in units of $K),$ and $c$ is its heat capacity $($with units $J{K}^{-1}).$ The

sphere behaves like a black$-$body and radiates a power per unit area $q=-T^4$ where $$ is

the Stefan$-$Boltzmann constant $($with units $W{m}^{-2}{K}^{-4}).$

$($a$)[5$pts$]$ The temperature $T(t)$ of the sphere is governed by the energy balance

$\frac{d}{dt}([$ thermal $],[$ energy $],[of$ the sphere $])=([$ total $],[$ radiated $],[$ power $])$

Derive an ODE for $T(t)$ in the form

$\frac{dT}{dt}=f(T,t)$

$($b$)7pts$ At time $t=0,$ the temperature of the sphere is $T_0.$ Find the analytical solution

of the ODE found in $($a$).$

$($c$)[7$pts$]$ Write pseudo code that implements the ODE found in $($a$)$ using the following

methods:

explicit Euler method

implicit Euler method

modified Euler method $($explicit$)$

Heun method $($explicit$)$

$($d$)[3$pts$]$ Explain the difference between implicit and explicit methods. What are advan$-$

tages and disadvantages of the two types of methods?

$($e$)[3$pts$]$ Explain the difference between single$-$step and multi$-$step integration methods.

List examples of explicit and implicit multistep methods.