An ordinary differential equation involves only one independent variable, a partial differential equation involves more than one. Examples of ordinary equations:
| d2y | + | 1 | dy | + y = 0; | d2x | + a | dy | + px + qy = 0. |
| dx2 | x | dx | dt2 | dt |
Examples of partial differential equations:
| x | dz | + y | dz | = nz; | d2u | + | d2u | + | d2u | = 0. |
| dx | dy | dx2 | dy2 | dz2 |
Equations, whether ordinary or partial, can
also be classified as linear or non-linear. A linear equation is a rational integral equation of the first degree in the dependent variable or variables and their derivatives. The equation
| x2 | d2y | + x | dy | + (x4 + 1)y = 0 |
| dx2 | dx |
is linear, but
| ( | dy | ) | 2 | = xy and y | dy | = x2 | ||
| dx | dx |
are non-linear. The order of an equation is the order of the highest derivative or differential which it contains. Of the three equations last written, the first is linear of the second order, the other two are of the first order and second degree. To integrate a differential equation or system of equations is to find a relation or relations among the variables, equivalent to the given equation or equations. Thus the integral of