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
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