The differential equation y′′(t) = y2(t) − 1
with boundary conditions y(0) = 0 and y(1) = 1 can be solved by using the initial conditions y(0) = 0 and y′(0) = s. By varing the parameter s and solving the equation numerically on [0, 1] we see that the value the numerical solution Y at 1 varies.
Solve the initial value problem for at least 40 values of s between 0 and 1. For all of these solutions determine the value of the numerical solution at 1 and save it an array, say e. Find the sign change of the vector e − 1( i.e. find the index for wchich the signs of e − 1 change. You can use fzero.m) and compute and plot the solution of the equation corresponding value of e.
Problem 2: Once you determined the value of s from the previous problem use MatLab to compute the definite integral
from 0 to 1 dy/sqrt(2(s + y^3/3 − y)).
please provide the coding