The general solution is the set of all solutions of this equation, also called the solution set. The special solution is one of all solutions of this equation, that is, an element in the solution set.
For example, the general solution produces y=kx (general solution) and y=2x (special solution).
For example:
If the solution of the differential equation contains any constant, and the number of any constant is the same as the order of the differential equation, such a solution is called the general solution of the differential equation! For example, y = x 2+c is the general solution of y'=x, because there is any constant c in y = x 2+c, y'=x is a first-order differential equation, and any constant is equal to order, so it is the general solution.
Y=c 1x+c2 is the general solution of y''=c 1, c 1 and c2 are two arbitrary constants that cannot be combined, and y'' is a second-order differential equation whose order is equal to the number of arbitrary constants, so it is the general solution.