Definition:Differential Equation/Solution

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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.