Q/7 and Q/5 P.I. =(cosx-sin x+e*)2 4Hence, the complete solution isy = Ce+e2|C, cos-1 1x++-(cosr-sinx+e*)2 4x+C3 sin-Ans.EXERCISE 3.22Solve the following differential equations :d'y1.+6y=sin 4xAns. C, cos ox+C, sin Jox -s-sin 4.xd'x2.dx+2+3x=sintAns. e"[Acos /2t + Bsin 2]-(co(cost-sint)4dtdr+2dtdx+5x=sin 2t, given that when t=D0, x = 3 anddt53cos 2t +343.di?55Ans. e17-sin 2t -(4cos 2t-sin 21)17dy4.dx-7 +6y = 2sin 3x, given that y = 1,dx-=0 when x = 0.de13Ans.e6x, 27e*+(7cos 3x-sin 3x)7575255. (D + 1) y = 2cos xAns. Cje+e2 | C, cos-2V31(-8sin 2x+ cos 2x)65(+C; sin-x+1+6. (D+ a) y = sin ax(A.M.I.E.T.E., June 2009) (Ans. C, cos ax+C, sin ax--COs ax2ax²COs axa²7. (D + 2a D + a) y = 8 cos axAns. (C +C,x+C; cos ax+ C, sin ax)-d'y8.+2y= sin 2xdx(A.M.I.E.T.E., Summer 2002)Ans. Ce+C,e 2*-.1(3cos 2x+sin 2.x)201[-sin 5x-12xcos x]489.+y=sin 3x cos 2xAns. C, cos x+C2 sin x+dxd'ydy10.2-3y = 2e2* +10 sin 3x given that y (0) = 2 and y'(0) 4dxdr29Ans.1212 cos 3x- 2sin 3x]d²dy+3+2y 4cos? x(R.G.P.V., Bhopal, I Semester, June 2007)11.dx?dx1(3 sin 2x- cos 2.x) +110Ans. Ce+C2e¯- 22xd²y12.dy+3y= cosx+x²dxdxsin -x) +

Answered: Image /qna-images/answer/5b687844-e03c-4c24-b062-c80e0e1e823a.jpg

75 25 75 5. (D+ 1) y = 2cos x %3D V3 Ans. Ce +e2" C2 cos- -x+C3 sin- (-8sin 2x+ cos 2x) 65 6. (D + a)y = sin ax (A.M.I.E.T.E., June 2009) (Ans. C cos ax+C, sin ax- cos ax 2a 7. (D + 2a D + a*) y = 8 cos ax Ans. (C +C,x+C; cos ax+C, sin ax)- os ax a'

75 25 75 5. (D+ 1) y = 2cos x %3D V3 Ans. Ce +e2" C2 cos- -x+C3 sin- (-8sin 2x+ cos 2x) 65 6. (D + a)y = sin ax (A.M.I.E.T.E., June 2009) (Ans. C cos ax+C, sin ax- cos ax 2a 7. (D + 2a D + a*) y = 8 cos ax Ans. (C +C,x+C; cos ax+C, sin ax)- os ax a'

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Q/7 and Q/5

P.I. =
(cosx-sin x+e*)
2 4
Hence, the complete solution is
y = Ce+e2|C, cos-
1 1
x++-(cosr-sinx+e*)
2 4
x+C3 sin-
Ans.
EXERCISE 3.22
Solve the following differential equations :
d'y
1.
+6y=sin 4x
Ans. C, cos ox+C, sin Jox -s
-sin 4.x
d'x
2.
dx
+2
+3x=sint
Ans. e"[Acos /2t + Bsin 2]-(co
(cost-sint)
4
dt
dr
+2
dt
dx
+5x=sin 2t, given that when t=D0, x = 3 and
dt
53
cos 2t +
34
3.
di?
55
Ans. e
17
-sin 2t -
(4cos 2t-sin 21)
17
dy
4.
dx
-7 +6y = 2sin 3x, given that y = 1,
dx
-=0 when x = 0.
de
13
Ans.
e6x, 27
e*+
(7cos 3x-sin 3x)
75
75
25
5. (D + 1) y = 2cos x
Ans. Cje+e2 | C, cos-
2
V3
1
(-8sin 2x+ cos 2x)
65
(+C; sin-
x+1+
6. (D+ a) y = sin ax
(A.M.I.E.T.E., June 2009) (Ans. C, cos ax+C, sin ax-
-COs ax
2a
x²
COs ax
a²
7. (D + 2a D + a) y = 8 cos ax
Ans. (C +C,x+C; cos ax+ C, sin ax)-
d'y
8.
+2y= sin 2x
dx
(A.M.I.E.T.E., Summer 2002)
Ans. Ce+C,e 2*-.
1
(3cos 2x+sin 2.x)
20
1
[-sin 5x-12xcos x]
48
9.
+y=sin 3x cos 2x
Ans. C, cos x+C2 sin x+
dx
d'y
dy
10.
2-3y = 2e2* +10 sin 3x given that y (0) = 2 and y'(0) 4
dx
dr
29
Ans.
12
1
2 cos 3x- 2sin 3x]
d²
dy
+3+2y 4cos? x
(R.G.P.V., Bhopal, I Semester, June 2007)
11.
dx?
dx
1
(3 sin 2x- cos 2.x) +1
10
Ans. Ce
+C2e¯
- 22x
d²y
12.
dy
+3y= cosx+x²
dx
dx
sin -
x) +

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