Second order nonlinear BVP shooting method

Question

I have BVP.

$y'' + 2y' - \frac{4}{x}y = 1$

$y'(\frac{1}{2}) = \frac{3}{2}; y(1) + y'(1) = 4$

$\frac{1}{2} \le x \le 1$

How can I solve it using shooting method? How can I get $y(a) = A $ and $y(b) = B$. And how cand I calculate $y(b,\alpha)$?

I will appreciate any help and links to theoretical material which can help.

Answer 1

As the differential equation is linear, the result depends linearly on the initial conditions. Thus solve once with $y'(\frac12)=0$ and once with $y'(\frac12)=1$ and find by linear interpolation the parameter $a$ where the second boundary condition is met for $y'(\frac12)=a$.