# Definition:Differential Equation/Solution/General Solution

## Definition

Let $\Phi$ be a differential equation.

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

## Also known as

The **general solution** of a differential equation $\Phi$ can also be referred to as **the solution** of $\Phi$, but beware of confusing this with the concept of **a solution** of $\Phi$.

The **general solution of a differential equation** can also be referred to as the **general solution to a differential equation**.

Some sources refer to this **general solution** as a **general integral**.

The term **solution set** is sometimes encountered.

## Also see

- Definition:Solution of Differential Equation
- Definition:Particular Solution of Differential Equation
- Definition:Singular Solution to Differential Equation

## Historical Note

The **general solution to a differential equation** was formerly known as the **complete integral**, or **complete integral equation**.

The Latin term used by Leonhard Paul Euler was **æquatio integralis completa**.

However, the term **integral equation** is now used to mean something completely different, and should not be used in this context.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.2$ Genesis of an Ordinary Differential Equation - 1956: E.L. Ince:
*Integration of Ordinary Differential Equations*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $2$. Integration - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**differential equation**