## You are here

Homehomogeneous linear differential equation

## Primary tabs

# homogeneous linear differential equation

The linear differential equation

$\displaystyle a_{n}(x)y^{{(n)}}+a_{{n-1}}(x)y^{{(n-1)}}+\ldots+a_{1}(x)y^{{% \prime}}+a_{0}(x)y\;=\;b(x)$ | (1) |

is called
homogeneous
iff $b(x)\equiv 0$. If $b(x)\not\equiv 0$,
the equation (1) is inhomogeneous.

If (1) is homogeneous,
then the sum of any solutions is a solution and any solution
multiplied by a constant is a solution.

The special case

$c_{n}x^{n}y^{{(n)}}+c_{{n-1}}x^{{n-1}}y^{{(n-1)}}+\ldots+c_{1}xy^{{\prime}}+c_% {0}y\;=\;0$ |

of (1), where the $c_{i}$’s are constants, can via the
substitution $x=e^{t}$ be transformed into a homogeneous^{}
linear differential equation of the same order but with
constant coefficients.

Major Section:

Reference

Type of Math Object:

Definition

Groups audience:

## Mathematics Subject Classification

34A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections