By F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester
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Additional info for A Modern Introduction to Differential Equations
This way of handling certain types of IVPs is common in physics and engineering texts. 4, the solution of the equation with y(−1) = 2, for example, is x t 2 − 2t + 7dt + 2 y(x) = −1 = = t3 − t 2 + 7t 3 − t=x x3 − x 2 + 7x − 3 t3 − t 2 + 7t 3 −25 3 +2= +2 t=−1 x3 31 − x 2 + 7x + . 1—that is, without using a deﬁnite integral formula. 2 Families of Solutions II Although we have seen examples of ﬁrst-order equations that have no solution or only one solution, in general we should expect a ﬁrst-order differential equation to have an inﬁnite set of solutions, described by a single parameter.
Rosenfeld, “Mathematical and Computer Modeling of Acute Normovolemic Hemodilution,” Transfusion 34 (1994): 176–179. Also see “Calculus in the Operating Room” by P. Toy and S. Wagon, Amer. Math. Monthly 102 (1995): 101. 1 Separable Equations b. 40, what is the patient’s volume of red blood cells at the end of the operation? 7. The volume V of water in a particular container is related to the depth h of the water by the = 16 4 − (h − 2)2 . If V = 0 when h = 0, ﬁnd V when h = 4. equation dV dh 8. The slope m of a curve is 0 where the curve crosses the y-axis, and dm = dx function of x.
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. ) After the analytic and qualitative treatments in this chapter, Chapter 3 will focus on some numerical solution methods concerned with approximating values of solutions. 1 1 In general, “elementary functions” are ﬁnite combinations of integer powers of the independent variable, roots, exponential functions, logarithmic functions, trigonometric functions, and inverse trigonometric functions.
A Modern Introduction to Differential Equations by F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester