nonlinear ODE shooting method using Newton $y'' = 2y'-2y+2$
with $y(0)=1 $ and $ y(\frac{\pi}{2})=2$
I have to solve this using shooting method (Newton).
First thing I need to do it replace the right boundary problem with a specified slope at left boundary. I am told in the question that $y'(0) = 0$. Then the note tells me to use Newton's method for $y(\frac{\pi}{2};0)-2$ ?
I am quite confused as to how to proceed and do this question. I cannot find any similar examples.
Thanks.

Since this is a second-order dynamical system you will need to compute the derivative at one or both boundaries. You will also need to rewrite the dynamics (differential equations) in first-order form.
The rough outline for a shooting method proceeds as follows:
1) guess the derivative (slope) at the start point
2) use an explicit integration scheme such as Euler' method, mid-point method, or 4th-order Runge-Kutta to simulate the system from the initial condition to the final one.
3) check the difference between the final value that your simulation achieved and the target final value. This is known as a defect.
4) update your initial guess for the slope and then loop back to (2). Typically this guess is updated by using the chain rule to compute a gradient estimate that describes how a change in the initial guess for the slope will affect the value at the final point. Google "Russ Tedrake, MIT, underactuated robotics, multiple shooting gradients" and you should be able to find a video lecture that explains how to compute these gradients. You can also approximate them numerically, which is easier in many cases but slower and less accurate.
If you are not constrained to using shooting methods then you might also look up direct collocation methods, which I have found to be superior to shooting methods for many problems. It also might be worth looking up multiple shooting, a simple extension of the algorithm that I described above, which has significantly better convergence properties.
