Deriving Implicit Euler Method Update Rule to be used in iterations

Question

Starting with the approximation $$y'(t) = \frac{y(t) - y(t - h)}h$$ arriving at an update rule of the form: $$y(t + h) = y(t) + hf(t + h, y(t + h))$$ Derive the implicit Euler update rule for this differential equation: $$y'(t) = 4e^{-t} y(t) - [y(t)]^2$$ with $$y(0) = 2$$ I really have no idea what to do. Any help would be greatly appreciated.

Answer 1

This is known as the Backward Euler method, which is a one-step method for numerically solving the equation $y' = f(t,y).$ In your case, the right hand side is: $$f(t,y) = 4e^{-t}y - y^2$$ and the initial value is given as $y(t_0) = 2 = y_0,$ where $t_0=2.$ Given step size $h>0,$ one usually denotes $y_n$ as the $n$-th iteration or equivalently $y_n$ is the approximation to the true value $y(t_n).$

For instance: $y_1 = y_0+hf(t_1, y_1)$ and in general: $$y_{n+1} = y_n + hf(t_{n+1}, y_{n+1}).$$ Now, the rest is just a matter of plug and chug...