General and particular solutions

A solution of a differential equation is a function that makes both sides equal when it and its derivatives are substituted. Its graph is called a solution curve (or integral curve).

Worked example: y = e^(2x) solves y″ − 3y′ + 2y = 0, because 4e^(2x) − 6e^(2x) + 2e^(2x) = 0. So does y = eˣ: 1 − 3 + 2 = 0.

A solution that contains arbitrary constants is the general solution. Giving the constants particular values produces a particular solution.

Worked example: y = A cos 3x + B sin 3x solves y″ + 9y = 0 for every A and B. The equation has order 2 and the general solution has two constants.

Conditions fix the constants. With y(0) = 2 and y′(0) = 6: A = 2 and 3B = 6, so B = 2 and the particular solution is y = 2 cos 3x + 2 sin 3x.

Worked example: the general solution of dy/dx = 2x is y = x² + C, a family of parabolas stacked vertically. The one through (1, 4) has 4 = 1 + C, so C = 3 and y = x² + 3.

Picture a first-order equation as a field of short slope marks: at each point (x, y) it gives the slope dy/dx. A solution curve follows the marks, and a starting point picks out one curve of the family.