= Solution
{c}
Introduce $v=u'$ and solve the initial-value system
$$
u'=v,\qquad v'=p(x)v+q(x)u-f(x,u),\qquad u(0)=\alpha,\quad v(0)=s.
$$
The 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:
$$
w'=z,\qquad z'=p(x)z+[q(x)-f_u(x,u)]w,\qquad w(0)=0,\quad z(0)=1.
$$
Then $R'(s)=w(1)$ 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.
Back to article page
