x and t are called unknowns—that is, unknown variables—because the objective of solving the equations is to find them. These equations are called simultaneous because they are to be solved together, at the same time, for values of x and t which will fit in both equations. The equations are called linear because the only powers of the unknowns that appear are the first power. Values that solve equations are said to satisfy them. It is easy to solve these two equations and find that x = 2 and t = 1 is their solution. But it is a long process to solve 10 linear simultaneous equations in 10 unknowns, and it is almost impossible (without using a mechanical brain) to solve 100 linear simultaneous equations in 100 unknowns.
derivative, integral,
differential equation, etc.
See the sections in [Chapter 5] entitled “Differential Equations,” “Physical Problems,” and “Solving Physical Problems.” There these ideas and, to some extent, also the following ideas were explained: formula, equation, function, differential function, instantaneous rate of change, interval, inverse, integrating. See also a textbook on calculus. If y is a function of x, then a mathematical symbol for the derivative of y with respect to x is Dₓ y, and a symbol for the integral of y with respect to x, is ∫y dx. An integral with given initial conditions ([see p. 83]) is a definite integral.
exponential
A famous mathematical function is the exponential. It equals the constant e raised to the x power, eˣ, where e equals 2.71828.... The exponential lies between the powers of 2 and the powers of 3. It can be computed from:
| eˣ = 1 + | x² | + | x³ | + . . . |
| 1 · 2 | 1 · 2 · 3 |
It is a solution of the differential equation Dₓy = y. See also a textbook on calculus. The exponential to the base 10 is 10ˣ.
logarithm
Another important mathematical function is the logarithm. It is written log x or logₑ x and can be computed from the two equations:
log uv = log u + log v