Analytical Solution to Problem III

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

wher D is an unknown constant. The boundary conditions are used to find the constants. Evaluating at x = 0 where y = 0 gives

0 + 0 = C 0 + D, or D = 0

Evaluating at x = 1 where y = 1 gives

This gives

This can be simplified to give

The plus sign must be chosen to satisfy the boundary condition at x = 0.

Take Home Message: This is an example of "separating variables and integrating" to find the analytic solution.