Definition:Differential Equation/Solution

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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

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 $\Phi$.


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 Poincaré.


Sources