_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: _x_² _x_³ _eˣ_ = 1 + _x_ + —————— + ————————— + ... 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_ _x_² _x_³ log(1 + _x_) = _x_ - —————— + —————— - ..., _x_² < 1 2 3 It is a solution of the differential equation _Dₓy_ = 1/_y_. If _y_ is the logarithm of _x_, then _x_ is the _antilogarithm_ of _y_. The _logarithm to the base 10_ of _x_, log₁₀ _x_, equals the _logarithm to the base e_ of _x_, logₑ _x_, divided by logₑ 10. See also textbooks on algebra and calculus. sine, cosine, tangent, antitangent These also are important mathematical functions. The _sine_ and _cosine_ are solutions of the differential equation _Dₓ_(_Dₓy_) =-_y_ and are written as sin _x_ and cos _x_. They can be computed from _x_³ _x_⁵ sin _x_ = _x_ - —————— + ————————— - ... 1·2·3 1·2·3·4·5