MATH 251 Lecture Notes - Ordinary Differential Equation, Partial Differential Equation, Separation Of Variables

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What is a differential equation? y = f (t). A differential equation is any equation containing one or more derivatives. The simplest differential equation, therefore, is just a usual integration problem. Comment: the solution of the above is, of course, the indefinite integral of f (t), y = f(t) + c, where f(t) is any antiderivative of f (t) and c is an arbitrary constant. Such a solution is called a general solution of the differential equation. It is really a set of infinitely many functions each differ others by one (or more) constant term and/or constant coefficients. Every differential equation, if it does have a solution, always has infinitely many functions satisfying it. All of these solutions, which differ from one another by one, or more, arbitrary constant / coefficient(s), are given by the formula of the general solution.

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