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Additional info for Advanced Engineering Math 9th Edition with Mathematica Computer Manual
12 we have p ϭ 4; hence W ϭ ce؊4x by integration from 0 to x, where c ϭ y1(0)y 2Ј(0) Ϫ y2(0)y 1Ј(0) ϭ 1 ⅐ Ϫ 0 ⅐ (Ϫ2) ϭ . 7. Nonhomogeneous ODEs. page 78 Purpose. We show that for getting a general solution y of a nonhomogeneous linear ODE we must find a general solution yh of the corresponding homogeneous ODE and then— this is our new task—any particular solution yp of the nonhomogeneous ODE, y ϭ yh ϩ yp. Main Content, Important Concepts General solution, particular solution Continuity of p, q, r suffices for existence and uniqueness.
This is a typical solution of a forced oscillation problem in the overdamped case. The general solution of the homogeneous ODE dies out, practically after some short time (theoretically never), and the transient solution goes over into a harmonic oscillation whose frequency is equal to that of the driving force (or electromotive force). 1 4. y ϭ A cos 3x ϩ B sin 3x ϩ _81 cos x ϩ _ 18 x sin 3x. An important point is that the Modification Rule applies to the second term on the right. Hence the best way seems to split yp additively, yp ϭ yp1 ϩ yp2, where yp1 ϭ K1 cos x ϩ M1 sin x, yp2 ϭ K2 x cos 3x ϩ M2 x sin 3x.
14. y1 ϭ x؊2, y2 ϭ x 2, W ϭ Ϫ4x؊1. Hence (2) gives ͵ ͵ yp ϭ Ϫx ؊2 x 2x؊4(x/4) dx ϩ x 2 x؊2x؊4(x/4) dx 1 ؊2 ϭ Ϫ_14 x؊2 ln ͉x͉ Ϫ _ 16 x . 16. y ϭ ux؊1/2 leads to u Љ ϩ u ϭ 0 by substitution. (This is a special case of the method of elimination of the first derivative, to be discussed in general in Prob. 5 on the Bessel equation. ) Hence a basis of solutions of the homogeneous ODE corresponding to the given ODE is y1 ϭ x؊1/2 cos x, y2 ϭ x؊1/2 sin x. The Wronskian is W ϭ x؊1 and r ϭ x؊1/2 cos x. Hence (2) gives ͵ sin x ͵ x (cos x) x (cos x) x dx cos x ͵sin x cos x dx ϩ x sin x ͵ cos x dx yp ϭ Ϫx؊1/2 cos x x؊1/2 (sin x) x؊1/2 (cos x) x dx ϩ x؊1/2 ϭ Ϫx؊1/2 ؊1/2 ؊1/2 ؊1/2 2 ϭ _12 x1/2 sin x.
Advanced Engineering Math 9th Edition with Mathematica Computer Manual by Erwin Kreyszig