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F&C sections 2.4, 4.3, 4.5, 4.6
A differential equation is an equation that express a relationship between some functions and their derivatives. Let's start with an analogy to algebraic equations. The simplest algebraic equation expresses a relation between a variable and a constant (a known number), x=a. The solution is trivial. The next level of complexity is a relation between a variable, its square, and a constant, ax2 + bx + c = 0. This equation is so common that it earned a special name, the quadratic equation, and we normally spend a considerable amount of time in basic algebra learning how to solve this equation. Further complexitiy is added by either including higher powers of x in the equation and/or including additional variables like y and z.
The simplest differential equation expresses a relation between the derivative of a function and a known function (of the differential), dx/dt = f(t). In this expression, x is to be regarded as a function, x(t). The solution is not quite trivial, but, the statement goes, "a solution exists", and this solution is x = òf(t)dt. A more complex situation exists if the differential equation involves the function x itself and/or higher derivatives of x. We will spend some time learning about the solutions to some of these equations as we look at how they appear in the context of mechanics.
© 25 March 1999 R. Harr