1.2. Can we write solutions explicitly? 5 For example, f (x) = sin(x)f 2 (x) and ut = xuxx are nonautonomous differential equations while f (x) = 9f 2 (x) and ut = 12uxx are autonomous. Since most of the nonlinear equations we will consider in this book will be autonomous, the following fact about solutions to such equations may prove useful: Theorem 1.3 If f(x1,...,xn) is a solution to an autonomous differential equation, then the function g(x1,...,xn) = f(x1 + c1,x2 + c2,...,xn + cn) obtained from it by adding constants ci to each of the variables is also a solution. Proof It is important to note that any partial derivative of g is equal to the corresponding partial derivative of f with the values of the variables shifted by the same constants. Moreover, in an autonomous equation, the variables appear only as arguments of the unknown function. Then, substituting g into the equation is the same as sub- stituting f into the equation but with different values of the variables. However, that f is a solution means that the equation is satisfied for all values of the variables and consequently is also satisfied after such a shift. Note that this is not true in general for nonautonomous equations. For instance, we noted that xex is a solution to the nonautonomous equation (1.5). However, you can check for yourself that (x + 4)ex+4 is not. 1.2 Can we write solutions explicitly? In the examples shown above, we were able to write formulas for exact solutions to the differential equations. In many cases, however, it is

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.