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9.1 - 9.1 Modeling with Differential Equations(Dated...

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9.1: Modeling with Differential Equations (Dated: November 8, 2011) Roughly speaking, a differential equation is an equation that contains an unknown function and some of its deriva- tives. The order of the equation is the order of the highest derivative that occurs in the equation. A function is called a solution of a differential equation if the equation is satisfied when the function and its deriva- tives are substituted into the equation. There are generally more than one solutions to a differential equation. Solving a differential equation means calculating all possible solu- tions of the equation. A representation of all possible solu- tions of a differential equation is called the general solution of the equation. If, in addition to a differential equation, a condition of the form y ( t 0 ) = y 0 is also given, then the problem of calculat- ing the solution of the differential equation that also satis- fies the initial condition y ( t 0 ) = y 0 is called an initial-value problem. Example 1 [Exercise 9.1.1 in the text] If y = x - 1 x , then y 0 = 1 + 1 x 2 , from which it follows that xy 0 + y = x 1 + 1 x 2 + x - 1 x
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