# IE13

From Example Problems

Transform the BVP to an integral equation:

First, integrate both sides from to 1 (so that the boundary condition
can be applied). This gives

Applying the condition and multiplying both sides by (-1) results in

Next, integrate both sides from 0 to :

or, applying the condition ,

The double integral may be simplified if we invert the order of integration. To do this, consider the region of integration for the double integral:

This same region, when the order of integration is reversed, looks like this:

We see that the region must be split into two pieces, and a separate integral written for each piece:

or after doing the trivial -integration,

The desired integral equation, then, is

Such an integral equation is often written in the form

where