# Definition:Differential Equation/Solution

## Definition

Let $\Phi$ be a differential equation defined on a domain $D$.

Let $\phi$ be a function which satisfies $\Phi$ on the whole of $D$.

Then $\phi$ 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

Let $\Phi$ be a differential equation.

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 $\Phi$.

### Weak Solution

A **weak solution** is a solution of a non-standard formulation of a **differential equation**.

## Also known as

Some sources refer to such a function $\phi$ as an **explicit solution** of $\Phi$.

## Examples

### Arbitrary Order $1$ ODE: $1$

Consider the real function defined as:

- $y = \ln x + C$

defined on the domain $x \in \R_{>0}$.

Then $\map f x$ is a solution to the first order ODE:

- $(1): y' = \dfrac 1 x$

defined on the domain $x \in \R_{>0}$.

### Arbitrary Order $1$ ODE: $2$

Consider the real function defined as:

- $y = \tan x - x$

defined on the domain $S := \set {x \in \R: x \ne \dfrac {\paren {2 n + 1} \pi} 2, n \in \Z}$.

Then $\map f x$ is a solution to the first order ODE:

- $(1): y' = \paren {x + y}^2$

when $x$ is restricted to $S$.

### Arbitrary Order $2$ ODE

Consider the real function defined as:

- $y = \map f x = \ln x + x$

defined on the domain $x \in \R_{>0}$.

Then $\map f x$ is a solution to the second order ODE:

- $(1): \quad x^2 y'' + 2 x y' + y = \ln x + 3 x + 1$

defined on the domain $x \in \R_{>0}$.

### Arbitrary Order $2$, Degree $3$ ODE

Consider the equation:

- $(1): \quad y = x^2$

where $x \in \R$.

Then $(1)$ is a solution to the second order ODE:

- $(2): \quad \paren {y''}^3 + \paren {y'}^2 - y - 3 x^2 - 8 = 0$

defined on the domain $x \in \R$.

### Equation which is Not a Solution

Consider the equation:

- $(1): \quad y = \sqrt {-\paren {1 + x^2} }$

where $x \in \R$.

Consider the first order ODE:

- $(2): \quad x + y y' = 0$

Then despite the fact that the formal substition for $y$ and $y'$ from $(1)$ into $(2)$ yields an identity, $(1)$ is not a solution to $(2)$.

### Absolute Value Function

Consider the real function defined as:

- $\map f x = \size x$

where $\size x$ is the absolute value function.

Then $\map f x$ cannot be the solution to a differential equation.

However, by suitably restricting $\map f x$ to a domain which does not include $x = 0$, there may well exist differential equations for which the resulting real function is a solution.

## 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 PoincarĂ©.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.2$ Genesis of an Ordinary Differential Equation - 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Definition $3.4$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions - 1978: Garrett Birkhoff and Gian-Carlo Rota:
*Ordinary Differential Equations*(3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction - 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**