Definition:Differential Equation/Solution/General Solution

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This article is complete as far as it goes, but it could do with expansion.
In particular: The concept of a "general solution" applies to all sorts of equations, e.g. $\cos x = \cos \alpha$ and so on, so the page Definition:General Solution needs to be a transclusion page not a redirect. A job for another time.
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Definition

Let $\Phi$ be a differential equation.

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



This article is complete as far as it goes, but it could do with expansion.
In particular: Include the case where $\Phi$ is actually a system of equations.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


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


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

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