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2001) Ans. 14 (2x + 3y) -9 log (14x + 21y + 22) = 49x + C 4. S. 1998) Ans. eX = y2 (x + C) 5. S. 1997) Ans. (X2 + y2 + 2) = 2e l /2 6. S. 1998) 33 A Textbook of Engineering Mathematics Volume - II 7. The equations of motion of a particle are given by dx dt + wy = 0, dy _ wx = 0, Find the path of the particle and so that it is a circle. 2009) Hint. y(t) = Cl cos wt + C2 sin wt x(t) = C2 cos wt - Cl sin wt so x2 + y2 = C~ + C~ = R 2 Objective Type of Questions Choose a correct answer from the four answers given in each of the following questions: 1.
P. P. of %(-;) (i cos ax - sin ax) 1 x =- - - cos ax 2 a 1. --:---:- sm ax = .. I. P. of = - x 2a 2 1 D +a . 2 e,ax -~ (~) (i cos ax - sin ax) . sIn ax 1 x . ---::---:-- cos ax = - sm ax .. D2 + a 2 2a Example 4. Solve (D2 + D + 1) Y = sin 2x Solution. Here the auxiliary equation is m 2 + m + 1 = 0 which gives m :. P. 1. = 2 21 sin 2x replacing D2 by - 22 (-2) +D+1 = -1- D-3 . 2x sm 1 (D - 3) (D + 3) 21 D -9 (D + 3) sin 2x (D + 3) sin 2x = ~ 21 -9 (D + 3) sin 2x = -~ (D + 3) sin 2x = -1 [D (sin 2x) + 3 sin 2x] 13 13 = -~ [2 cos 2x + 3 sin 2x] Since D means differentiation with respect to x 13 :.
P. I. A particular integral of the differential equation f(D) y = Q is given by _1_ Q f(D) Methods of finding Particular integral (A) Case I. , when Q is of the form of eax, where a is any constant and f(a) :1; 0 we know that 41 A Textbook of Engineering Mathematics Volume - II D2 (eax) = a2 eax (e ax) = a 3 e ax Dn (e ax) = an e ax :. I. : f(a) is constant e ax f(D) e ax = _1_ f(a) eax if f(a) "# 0 , Case II. I. I. = - - e ax f (D) = eax x2 - - if f" (a) "# 0 f" (a)' Example 1. Solve (D2 - 2D + 5) Y = e- X 42 Linear Differential Equations with Constant Coefficients and Applications Solution.
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