# Boundary Value Problems/Introduction to BVPs

## Objective

[edit | edit source]Introduce Boundary value problems for a single independent variable.

## Approach

[edit | edit source]- 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.

## Initial Value Problems

[edit | edit source]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

## Two point BVPs for an ODE

[edit | edit source]Begin with second order DEs, , with conditions on the solution at and .

with and on the interval

## Example

[edit | edit source]with and on the interval

See the wikipedia topic