Test the understanding!

Derive Forward Euler, Backward Euler, and Crank-Nicolson schemes for Newton's law of cooling: $$T' = -k(T-T_s),\quad T(0)=T_0,\ t\in (0,t_{\mbox{end}}]$$

Physical quantities:

• $T(t)$: temperature of an object at time $t$
• $k$: parameter expressing heat loss to the surroundings
• $T_s$: temperature of the surroundings
• $T_0$: initial temperature