By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester
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Extra resources for A Modern Introduction to Differential Equations
Such an equation can be described in terms of its order, the order of the highest derivative of the unknown function in the equation. Differential equations can also be classiﬁed as either linear or nonlinear. Linear equations can be written in the form an (x)y (n) + an−1 (x)y (n−1) + · · · + a2 (x)y + a1 (x)y + a0 (x)y = f (x) where each coefﬁcient function ai (x) depends on x alone and doesn’t involve y or any of its derivatives. Nonlinear equations usually contain products, quotients, or more elaborate combinations of the unknown function and its derivatives.
Next, there is a qualitative way of viewing differential equations. This is a neat geometrical way of studying the behavior of solutions without actually solving the differential equation. The idea is to examine certain pictures or graphs derived from differential equations. Although we can do some of this work by hand, computer algebra systems and many graphing calculators can produce these graphs, and you’ll be expected to use technology when appropriate. There are also specialized programs just for doing this sort of thing.
Assuming constant acceleration, we ask how far the car travels before it reaches 60 mph. 5-second interval. We know the acceleration can be d2 s described as a(t) = dt 2 , which in this problem equals some constant C; and we know that s(0) = s (0) = 0—that is, our initial position is considered 0, and the velocity when we ﬁrst put our foot on the gas pedal is also 0. 5 seconds, is 60 mph. Thus, we have a second-order differential d2 s equation dt 2 = C, initial conditions, and some boundary conditions, and we must solve for the unknown function s(t).