In reaction $2AB,$ the concentration of $A(c)$ changes with time following second$-$order kinetic:

$\frac{dc}{dt}=-0\mathrm{.}2c^2$

$($a$)$ Write down the midpoint scheme for solving the concentration of $A$ at $t^{n+1}$ i$.$e$.c(t^{n+1}).$

$($b$)$ How do you compute $c(t^{n+1})$ from $c(t^n)$ using the backward Euler scheme? Write $c(t^{n+1})$ as a function of only $c(t^n)$ and $t.$