Shooting Method

This method is widely used. It consists of transforming the boundary value problem to an initial value problem.

We consider a second order linear equation

(7.9 1a)

(7.9 1b,c)

We convert the boundary value problem to an initial value problem by dropping the boundary condition given by Equation (7.9 1c) and replacing it by imposing a condition at x=a. We assume that

(7.9 2)

Equations (7.9 1a,b, 2) form an initial value problem and we can integrate the differential equation using one of the formulas given in the previous section. Since ? ₁ is a guessed value of y'(a), it is unlikely that the boundary condition given by Equation (7.9 1c) will be satisfied. We now guess another value of y'(a) (say ? ₂) and we proceed with the integration. Unless we are lucky, Equation (7.9 1c) will still not be satisfied.

In the linear case, from these two values of ? ( ? ₁ and ? ₂) it is possible to obtain the exact value of y ?(a). Let y ₁ be the solution obtained by using ? ₁ and y ₂ be the solution with ? ₂, as illustrated in Figure...