A differential equation is an equation that includes differentials or derivatives. A solution to a differential equation is any function that satisfies the given equation. Thus f (x) = sin x is a solution to the differential equation y′′ + y = 0, because if y = sin x, then y′ = cos x and y′′ = - sin x, and hence y′′ + y = sin x - sin x = 0. Differential equations are encountered in the study of problems in both pure and applied mathematics, in the sciences, in engineering, and in business and the social sciences.
With the choices on the Solve ODE submenu you will be able to find closed-form solutions to many differential equations. The solution is generally returned as an equation in y(x) and x (or whatever variables were specified) with any arbitrary constants represented as C1, C2, ..., Cn
.
To solve a differential equation
Solve ODE + Exact
= xy, Exact solution is : y
x
= e
x2C1
To check this result, define
yx
= e
x2C1.
Replace y by y(x) in the differential equation and evaluate both sides.
Evaluate
= xe
x2C1
xyx
= xe
x2C1
For any given number C1, the solution describes a curve. Since C1 may, in general, take on infinitely many values there are an infinity of solution curves, or a one-parameter family of solution curves for this equation.