| d2z | = g is z = ½gt2 + At + B, |
| dt2 |
where A and B are arbitrary constants. An ordinary equation of the nth order with one dependent variable has exactly n arbitrary constants in its complete integral, or solution. In a practical problem the arbitrary constants are determined by the initial, or boundary, conditions. The solution of d2z/dt2 = g, e.g. is completely determinate if the values of z and dz/dt when t = 0 are given. The solution of partial equations may involve arbitrary functions, which become definite when proper initial or boundary conditions are assigned. Thus the equation du/dx = du/dt has for its complete solution u = φ(x + t), where φ may be a function of any form whatever; if now we are given that, when t = 0, u = a given function f(x), we obtain f(x) = φ(x), so that the solution required is u = f(x + t). Certain ordinary linear equations of the second order are specially important, both from the beauty of their theory and from their usefulness in Mathematical Physics. Some of these equations are: Bessel's equation, Legendre's equation, the hypergeometric equation, Mathieu's equation, Lamé's equation. Linear partial equations of the second order are fundamental in Physics. Such are: Laplace's equation,
| d2V | + | d2V | + | d2V | = 0; |
| dx2 | dy2 | dz2 |
the wave equation,
| d2V | = c2 | ( | d2V | + | d2V | + | d2V | ) | ; | ||
| dt2 | dx2 | dy2 | dz2 |
the equation of conduction of heat,
| dV | = k | ( | d2V | + | d2V | + | d2V | ) | . | ||
| dt | dx2 | dy2 | dz2 |
These involve one dependent variable only. Equations with several dependent variables occur in Elasticity, Electrodynamics, and Hydrodynamics. A notable feature of the hydrodynamical equations is that they are not linear.
No general rules exist enabling us to deal with a differential equation taken at random, and only a few types have been completely solved. Of soluble equations, the most important are those which are linear with constant coefficients.
Example 1. d2x/dt2 - 7dx/dt + 12 = 0. To solve this, try x = emt. We find emt(m2 - 7m + 12) = 0. Thus m = 3 or 4. It is now easy to show that x = Ae3t + Be4t is a solution, where A and B are arbitrary constants. This is the general solution. We can determine A and B if the values of x and dx/dt are given for a definite value of t, say t = 0.