# Predictor-corrector method applicaton

As an exercise I have to solve the easy ODE $$y' = \alpha y, \quad \alpha \in \mathbb{R}$$

with the predictor-corrector method where the predictor method is the explicit Euler method and the corrector method is the implicit trapezodial rule $$y_{j+1} = y_j + \frac{h}{2}(f(t_j, y_j) + f(t_{j+1}, y_{j+1})).$$ For the exercise, I have to determine $y^{(0)}_{j+1}$ after doing one step with the predictor as well as $$y^{(k)}_{j+1} = y_j + \frac{h}{2}(f(t_j, y_j) + f(t_{j+1}, y_{j+1}^{(k-1)}) \qquad (A).$$ Furthermore, I have to come up with a formal expression $$y_{j+1}^{(k)} = y_j + h\phi^{(k)}(t_j, y_j) \qquad (B).$$ The problem is that it's my first encounter with predictor-corrector methods and I do not have a clue where to start. There are several things that I don't understand:

First, how can I determine $y_{j+1}^{(0)}$ with the explicit Euler without having any type of actual values as to which step to iterate to, values for $t$ etc.? Second, how to derive the expressions in (A) and (B)? As I said, I have no previous experience with predictor-corrector methods. Although I read some theory beforehand, it does not seem to help me solve relevant exercises.

Hint: Insert the given function at the values indicated in the formulas: $$y^{(0)}_{j+1}=y_j+h·αy_j.$$ and $$y^{(k)}_{j+1}=y_j+h·\frac{αy_j+αy^{(k-1)}_{j+1}}2.$$