By Albert L. Rabenstein
Designed to be used by way of sophomore engineering or junior actual technological know-how majors, this article is acceptable for an introductory path in linear algebra and differential equations or a path in differential equations with a linear algebra prerequisite. this article includes distinct insurance of utilized subject matters and comprises theorems particularly appropriate to engineering scholars. there's a new bankruptcy on "Stability and the part Plane", nearly three hundred new difficulties extra all through and a number of other uncomplicated courses on numerical suggestions of differential equations are incorporated
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Additional resources for Elementary Differential Equations with Linear Algebra
Y — 2(sin x) y = — 2y3/2 sin x -y + 2(x2-\)y3=0 x+1 In Exercises 22-24 find a new dependent variable such that the equation becomes linear in that variable. Then solve the equation. 21. 2y+ 22. xeyy - ey = 3x2 23. ) 1 2 2 y +1 JC JC -T_y+-tan-1j>=- 24. 30) describes a family of parabolas. 2. Through every point (x 0 , y0) in the plane, except those points on the y-axis, there passes exactly one curve of the family. For if we specify (*o> yo) with x0 Ö 0, then c is determined by the condition y0 = cxl or The slope of the curve through the point (x, y) is y' = 2cx.
Using the data of Exercise 1, find the population as a function of time, assuming that the model dN/dt = kN1/2 applies. 3. (a) Find a formula for N(t) if N(0) = N0 and dN , where a is a positive constant, á Ö 1. (b) If 0 < a < 1 in part (a), what happens to N(t) as t becomes infinite? (c) If a > 1 in part (a), show that N(t) becomes infinite after a finite time. 9 Economic Models 41 4. 41) of Eq. 40). 5. Initially the population of a species is 5000. After 10 days it is 8000. After a very long time the population stabilizes at 15000.
34) The constants c and k can now be obtained from the conditions x(0) = x o , x(T) = ix 0 . 35) Here x0 is the amount of undecayed substance present at t = 0 and T is the half-life. Putting t = 0 in Eq. 34), we see that x0 = ce° or c = x0. Thus x(t) = x0e~kt. 6 Radioactive Decay 35 from which it follows that ekT = 2 and kT = In 2. Thus T and k are related as follows: T = I In 2, k = - In 2. Notice that T does not depend on x0, the amount of substance present initially. If T is known from experimentation, then k is found from the last formula.