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Unformatted text preview: First-order ordinary differential equations Before we get involved trying to understand partial differential equations (PDEs), well review the highlights of the theory of ordinary differential equations (ODEs). Well do this in such a way that we can begin to anticipate some of the methods well be using on PDEs later. An ordinary differential equation gives a relationship between a function of one independent variable, say y ( x ), its derivatives of various orders y ( x ), y 00 ( x ) etc. and the independent variable x . The order of the equation is the order of the highest derivative that appears. So a first-order differential equation can always be put into the form: F ( x, y, y ) = 0 . In general, it is possible to find solutions to such ODEs, and there is usually one and only one solution whose graph passes through a given point ( a, b ). In other words, there is one and only one solution of the initial-value problem : F ( x, y, y ) = 0 y ( a ) = b. At this level of generality, its impossible to say much more. But there are several special types of first-order ODEs for which solution techniques are known (i.e., the separable, linear, homogeneous and exact equations from Math 240). Well review the first two kinds, since we wont need the other two. A first-order differential equation is called separable if it can be put in the form: y = f ( x ) g ( y ) , so that you can separate the variables as dy g ( y ) = f ( x ) dx and then integrate both sides to get at least y as an implicitly-defined function of x . The constant of integration is then chosen so that the graph of the solution passes through the specified point in the initial-value problem. For instance, to solve y = xy y (0) = 1 , we separate: dy y = xdx and integrate: Z dy y = Z xdx 2 pde notes i to get ln y = x 2 2 + C. If y = 1 when x = 0, then we must have C = 0 and we can solve the resulting equation ln y = 1 2 x 2 to get y = e x 2 / 2 as the solution of the initial-value problem. A slight word of caution: there will be a technique in PDE called separation of variables. It has nothing to do with the kind of separation for first-order ODEs. Linearity is an important concept in many parts of mathematics. In the theory of differential equations (both ordinary and partial), we often think of the set of (differentiable) functions as comprising a vector space, since one can add two functions together to get another function, and one can multiply a function by a constant to get another function, just as one does with ordinary vectors. The addition and scalar multiplication of functions satisfies all the vector space axioms, so it is reasonable to think of functions as vectors....
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