General solution and particular solution

Builds on Solution of an ordinary differential equation Ordinary differential equation

Summary

The general solution of an ordinary differential equation of order $n$ is a family of solutions that depends on $n$ arbitrary constants $C_1, \dots, C_n$ . In the usual case, each solution of the equation is a member of the family, with the possible exception of singular solutions. A particular solution is one solution of the equation. It is often the member of the general solution for one choice of the constants.

Remarks

The general solution of $y'' + y = 0$ is $y = C_1 \cos x + C_2 \sin x$ . The choice $C_1 = 1$ , $C_2 = 0$ gives the particular solution $y = \cos x$ . An initial value problem fixes the constants. For an equation of order $n$ , one needs $n$ initial conditions. For a linear equation, the general solution of the inhomogeneous equation is one particular solution plus the general solution of the homogeneous equation.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
General solution and particular solution concept

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Tag	`0SKYFQAFYD`
URI	`m2://0SKYFQAFYD`
Kind	concept
Writing	W0 stub
MSC	34A05
Uses	Solution of an ordinary differential equation, Ordinary differential equation

General solution and particular solution

Summary

Remarks

Reading path

Where this leads

Further reading

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