$\frac{d^2 y}{ d x^2} = 4(y + \cosh (1))$ ,$ 0

Question

I want to use Runge kutta 4th order to solve it, which means l have to reduce it to first order, it seems impossible because it is a boundary value problem. My question is how can I do it without going through the shooting method.

Answer 1

If you want to ignore the obvious exact solution and compute the numeric one "by the book", you will need two forward integrations, with $(y_0(0),y_0'(0))=(1,0)$ and then with $(y_1(0),y_1'(0))=(1,1)$. As the problem is linear, any solution is obtained as $$ y(t)=y_0(t)+y'(0)\,(y_1(t)-y_0(t)). $$ This allows you to compute the necessary initial slope by solving this equation at $t=1$ for $y'(0)$.

Answer 2

Hint: $\frac{d^2y}{dx^2} = y' \frac{dy'}{dy}$