##### Exercise 4.12: Shooting algorithm.

This exercise illustrates the shooting algorithm for solving

Geometrically, we seek a function that satisfies the differential equation and whose graph passes through the points (a,A) and (b,B). Our approach is to determine , then we would have an initial value problem and RKSUITE could be used to solve it. So, let and the task is to find so that the resulting solution, denoted by , satisfies . We seek a zero of the function,

Step 1: Choose and solve the differential equation with initial conditions and ; denote the resulting solutions by . If

set and stop. The solution is .

Step 2: Choose and solve the differential equation with and ; denote the solution by . If

set and stop. The solution is .

Step 3. Calculate the values of s for which :

Note that, since our differential equation is linear, is a linear function of s.

Step 4. Solve the differential equations with , to get the desired solution.

The process is illustrated graphically in Figure 6.

Use the shooting method to solve the following boundary value problems. Plot your solutions: