# Definition:Linear Second Order ODE with Constant Coefficients

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## Definition

A **linear second order ODE with constant coefficients** is a second order ODE which can be manipulated into the form:

- $y'' + p y' + q y = \map R x$

where:

Thus it is a linear second order ODE:

- $y'' + \map P x y' + \map Q x y = \map R x$

where $\map P x$ and $\map Q x$ are constant functions.

## Also known as

The word ordering may change, for example:

**constant coefficient linear second order ODE**

Abbreviations can be used:

**constant coefficient LSOODE**

and so on.

## Also presented as

Such an equation can also be presented in the form:

- $\dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$

or:

- $\paren {D^2 + p D + q} y = \map R x$

## Also see

- Results about
**constant coefficient linear second order ODEs**can be found**here**.

## Historical Note

The first methods for solving linear second order ODEs with constant coefficients were devised by Leonhard Paul Euler.

The case where the auxiliary equation has repeated roots was addressed by Jean le Rond d'Alembert.

Rehuel Lobatto ($1837$) and George Boole ($1859$) worked on refining the symbolical methods.

## Sources

- 1958: G.E.H. Reuter:
*Elementary Differential Equations & Operators*... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 2$. The second order equation: $\S 2.1$ The reduced equation - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 18$: Basic Differential Equations and Solutions: $18.8$: Linear, nonhomogeneous second order equation - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**linear differential equation with constant coefficients**