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**Extra resources for Introduction to Ordinary Differential Equations. Academic Press International Edition**

**Sample text**

10 Linear Differential Equations EXERCISES 1. Find the general solution if x is restricted to the interval (0, + oo). (a) (b) (c) (d) (e) x2y" - 3xy' - 12y = 0 2x2y" + 3xy' - y = 0 j c 2 / - 3JC/ + 4J> = 0 4x2y" +y = 0 x2y" + xyf + 9y = 0 (f) (g) (h) (i) x2y" + 3xy' + y = 0 x2y" + 2xy' + 2y = 0 x2y'" + 2 χ 2 / - * / + y = 0 JC 3 /" + 4x2y" + 6 x / + 4j> = 0 2. Find the solutions of the initial value problem on the interval (0, +oo): (a) x2y" - 2xy' + 2y = 0, (b) x2y" + 5xy' + 4>; = 0, (c) J C 2 / - 3xy* + 5^ = 0, ><2) = 3, j ( l ) = 2, XI) = 2, /(2) = 1 y\\) = - 3 /(l) = 0 3.

Consider the situation of Problem 3, but with the effect of damping con sidered (c Φ 0). Assume that c2 < 4mk, and let c 2m a =■ JAkm 2m Find the position of the body as a function of time. Is there any significant change in the motion in the case when y = ω? 5. A body falls from rest from a point above the earth. If air resistance is neglected, show that the body falls the distance x = W1 in time /. 6. A body is thrown vertically upward from the surface of the earth with velocity v0. ) 7. A body is thrown vertically upward from the surface of the earth with velocity v0.

4. Use the result of Problem 3 to obtain particular solutions of the equations in Problems 1(a) and 1(b). 5. If the constant a is a root, of multiplicity m, of the polynomial equation P(r) = 0, then P(r) = Q(r)(r - a)m, where Q(a) Φ 0. Verify that the function v = — - — xmeax m\Q(a) is a solution of the differential equation P(D)y = Aeax. 6. Use the result of Problem 5 to find particular solutions of the equations in Problems 1(c) and 1(d). 7. If the function yp(x) is a solution of the equation P(D)y = Aeiax (A is real), then the real and imaginary parts of yp(x) are real solutions of the equations P(D)y = A cos ax, P(D)y = A sin ax, respectively.