Analytic Solution to Problem II.
The problem to be solved is
We can solve this equation easily by integrating twice. The first integration gives
where C is an unknown constant. The second integration gives
where D is an unknown constant. The boundary conditions are used to find the constants. Evaluating at x = 0 where y = 0 gives
0 = 0 + 0 + C 0 + D, or D = 0
Evaluating at x = 1 where y = 1 gives
The final result is
The exact solution for y(0.5) is 0.5885416.
Take Home Message: This is an example of "separating variables and integrating" to find the analytic solution.