How to combine both Secant and Runge-Kutta methods and make it to the two-point boundary-value.
Construct a computer program that uses both the secant method and the Runge-Kutta method to obtain a numerical solution to the two-point boundary-value problem:
x' = f(t,x) = x + 0.09 x 2 + cos(10 t) differential equation
x(0) + x(1) - 3.0 = 0 boundary condition
Starting with the initial guesses 0.7 and 1.0 for the (unknown) initial value, x(0), obtain an approximation to x(0) {for the final solution, x(t)} such that the boundary condition is satisfied to within a tolerance of 10-4.
Use a fixed stepsize of 0.025 (i.e., take 40 steps each time you integrate the differential equation from t=0 to t=1).
Write your program so that the output shows the values of x(0), x(1), and x(0)+x(1)-3 (the error in satisfying the boundary condition) at the end of each iteration of the secant method.
After the last iteration of the secant method, re-integrate from t=0 to t=1 and print out the solution for x(t) over the range [0,1].
