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Example Questions
Example Question #1 : Second Order Boundary Value Problems
Find the solutions to the second order boundary-value problem. ,
,
.
There are no solutions to the boundary value problem.
The characteristic equation of is
, with solutions of
. Thus, the general solution to the homogeneous problem is
. Plugging in our conditions, we find that
, so that
. Plugging in our second condition, we find that
and that
.
Thus, the final solution is .
Example Question #1 : Second Order Boundary Value Problems
Find the solutions to the second order boundary-value problem. ,
,
.
There are no solutions to the boundary value problem.
There are no solutions to the boundary value problem.
The characteristic equation of is
with solutions of
. This tells us that the solution to the homogeneous equation is
. Plugging in our conditions, we find that
so that
. Plugging in our second condition, we have
which is obviously false.
This problem demonstrates the important distinction between initial value problems and boundary value problems: Boundary value problems don't always have solutions. This is one such case, as we can't find that satisfy our conditions.
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