General solution and particular solution

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.

Explanations

L1 human

The general solution is the list of all answers, written with free constants. A particular solution is one answer from the list. For $y' = 2x$, the general solution is $y = x^2 + C$, and $y = x^2 + 5$ is a particular solution.