 Solution

Introduce $v=u^{\prime }$ and solve the initial-value systemThe unknown initial slope $s$ is the shooting method parameter. Integrate to one and solve the scalar residual equation $R(s)=u(1;s)-\beta =0$, using a secant iteration or a bracketed root finder when suitable data are available.

For a Newton update, integrate the sensitivity equations along the current shot:Then $R^{\prime }(s)=w(1)$ and the update is $s\leftarrow s-R(s)/w(1)$ when this derivative is nonzero. After a root has been found, the corresponding shot satisfies both endpoint conditions. This formulation does not presume that a nonlinear boundary-value problem necessarily has a unique solution.