1. Differential Equations 3 whether x3 + x2 and 2x2 are the same function. Of course, they are not because when you plug x = 2 into each side you do not get the same value so they must be different functions (even though they do have the same values for a few different inputs). More generally, differential equations can involve functions of more than one variable and derivatives higher than just the first derivative as well, as illustrated in the following example. Example 1.2 For what value(s) of the parameters λ and μ is the function f(x, y) = λ sin(y) + μx a solution to the nonlinear differential equation2 (f + fyy)(fx + fy − 1) = 0? (1.6) Solution Simply substituting the given definition for f into the expression for the equation gives us (μx) (μ + λ cos(y) − 1) = 0. Now, a product of numbers is zero only if one of the two factors is equal to zero, so we would need either μx = 0 or μ + λ cos(y) − 1 = 0. The first one is only true if μ = 0. (Note that we do not list x = 0 as a possibility since we are looking for the left and right side to be equal as functions of x, as explained before.) The second factor is equal to zero (as a function of y) only if μ = 1 and λ = 0. We can therefore conclude that this function f is a solution to the equation when μ = 0 (for any value of λ) and also when μ = 1 and λ = 0. 2 This makes use of the standard notation for partial differentiation in which fxx denotes the second derivative with respect to x of the function f.

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