Chapter Notes (2)

17 2 differential equations dy d x e c e x and since

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Unformatted text preview: Differential Equations dy d x = (e + C ) = e x and since the right side is just y = e x + C , the differential dx dx equation is not satisfied except when C = 0 . However, if we instead consider y = Ce x then we dy d find that = (Ce x ) = Ce x = y , so that these functions are solutions of the differential dx dx equation.! Here we find that dy in dx terms of either the independent or dependent variable or both. Second, the equation will usually have an infinite number of solutions expressible by varying some arbitrary constant. This family of solutions is called the general solution of the differential equation. To single out a unique solution we must specify some additional information, as in the next example. These two examples illustrate two characteristics of ODEs. First, the equation will express Example 2.3: Find the solution of dy = y that satisfies y (0) = 10 . dx Solution: We use the general solution y = Ce x found in Example 2.2 (a systematic method for finding these solutions will be discussed in the next section). Since when x = 0 we are supposed to have y = 10 , we can substitute this information into the general solution to determine C : 10 = y (0) = Ce0 = C . Thus, the specific solution we want is y = 10e x .! The data consisting of a differential equation together with a numerical value of the unknown function is called an initial value problem. Quite often, as in Example 2.3, the function value will be the value of y at x = 0 , but a value at some other x value may be given instead. This piece of information is referred to as the initial condition. For most problems there is a unique solution of the differential equation that satisfies the given initial condition. 2.2 Separation of Variables At this point the only type of ODE we can solve is one of the form y′ = f ( x) , provided we can compute the antiderivative of f ( x) . In this section we develop a technique that can be used to solve many differential equations. The method applies to what are called separable equations those for which the right side can be factored into a product or quotient of expressions, each of which involves only the independent or dependent variable. 18 2 D...
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This document was uploaded on 04/06/2014.

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