Introduce Boundary value problems for a single independent variable.
- What is a Boundary Value problem?
- Solution of a Boundary Value Problem is directly related to solution of an Initial Value Problem. So let's review the material on IVPs first and then make the connection to BVPs.
- Details of solving a two point BVP.
For a single independent variable
in an interval
, an initial value problem consists of an ordinary differential equation including one or more derivatives of the dependent variable,
,
and
additional equations specifying conditions on the solution and the derivatives at a point
, ...,
,
Example:
The differential equation is
(First order differential equation.) and the initial condition at
is given as
.
Solution:
.
When,
and
Get out a piece of paper and try to solve the following IVP in a manner similar to the preceding example:
and the initial condition at
is given as
.
Once you have an answer (or are stuck) check your solution here. Click here for the solution: IVP-student-1
A second order ODE example:
The differential equation is
(Second order differential equation.) and the two initial conditions at
given as
.
Solution:
Assume the solution has the form
The characteristic polynomial. Solve for "r".
See the Wikipedia link for more on Initial Value Problems
Begin with second order DEs,
, with conditions on the solution at
and
.
with
and
on the interval
with
and
on the interval
See the wikipedia topic
Boundary Value Problems