# Boundary Value Problems/Review: ODEs

Return to Boundary Value Problems

For more of an introduction to ODEs you are referred to Boundary Value Problems/BVP-Ordinary-Differential-Equations and for examples please look at Examples of ordinary differential equations

## First order differential equation:[edit | edit source]

Let the equation

is a linear ordinary (single independent variable) differential equation of first order (first derivative). If on the equation is called "homogeneous" differential equation. Otherwise it is called nonhomogeneous otherwise.

Examples of homogeneous and non-homogeneous:

where is a constant coefficient.

where is a coefficient that increases as increases.

where .

## Solving homogeneous ODE with constant coefficients: Separable case[edit | edit source]

Solving

where is the integration constant.

After integration there is a constant, an unknown. This will happen each time you integrate. To determine this unknown for a particular application you will need another piece of information. Typically it is another equation that requires . This condition with the above differential equation defines an **Initial Value Problem**. For this problem the value of is determined by using an intial condition such as . Then and making the substitution provides the solution

## Student work[edit | edit source]

Solve the following IVPs:

- Differential eq: where for all . Initial value:

Click here for solution: Boundary Value Problems/BVP-solution-1

- Differential eq: where for all . Initial value:

Click here for solution: Boundary Value Problems/BVP-solution-2

### Pre-class work:[edit | edit source]

Complete each of the following:

- Read the following:
- Pages 1-10 of Powers
- Lecture notes (web page)
- (View) Associated Lecture Video

- Work the following problems Boundary Value Problems/BVP-Exercise 1-1:Problems and Solutions