# Definition:Differential Equation/Solution

## Definition

Let $\Phi$ be a differential equation.

Any function $\phi$ which satisfies $\Phi$ is known as **a solution** of $\Phi$.

Note that, in general, there may be more than one **solution** to a given differential equation.

On the other hand, there may be none at all.

### General Solution

The **general solution** of $\Phi$ is the set of *all* functions $\phi$ that satisfy $\Phi$.

### Particular Solution

Let $S$ denote the solution set of $\Phi$.

A **particular solution** of $\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\Phi$.

## Historical Note

The original name for a **solution to a differential equation** that Jacob Bernoulli used in $1689$ was **integral**.

Leonhard Paul Euler used the term **particular integral** in his *Institutiones Calculi Integralis* of $1768$, but the latter term has more recently taken on a more precise definition.

The term **solution** was first used by Joseph Louis Lagrange in $1774$.

This terminology became established under the influence of Jules Henri Poincaré.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.2$ Genesis of an Ordinary Differential Equation - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.2$: General Remarks on Solutions - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**differential equation**