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Unformatted text preview: D. Definite Integral Solutions You will find in your other subjects that solutions to ordinary differential equations (ODEs) are often written as definite integrals, rather than as indefinite integrals. This is particularly true when initial conditions are given, i.e., an initial-value problem (IVP) is being solved. It is important to understand the relation between the two forms for the solution. As a simple example, consider the IVP (1) y = 6 x 2 , y (1) = 5 . Using the usual indefinite integral to solve it, we get y = 2 x 3 + c , and by substituting x = 1 , y = 5, we find that c = 3. Thus the solution is (2) y = 2 x 3 + 3 However, we can also write down this answer in another form, as a definite integral Z x (3) y = 5 + 6 t 2 dt . 1 Indeed, if you evaluate the definite integral, you will see right away that the solution (3) is the same as the solution (2). But that is not the point. Even before actually integrating, you can see that (3) solves the IVP (1). For, according to the Second...
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