Page 91 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

x
(b) c e + c e − 7x 42. Two linearly independent solutions of
2
1
the differential equation
11
x
1
−
x
7x
(c) c e + c e y − 2y + y = 0 are y = e and
1
1 2
y = xe . Then a particular solution
x
2
−
x
(d) c e + c e − 7x of y − 2y + = e x sin x is
11
1
y
2
1
40. The general solution of the (a) y cos x + y (sin x x cos ) x
−
differential equation 1 2
( cos x −
( ) 8y x =
x −
1
x
y 11 ( ) 4y x + ( ) 10 cos x (b) y + sin x + y x sin ) x
e
2
1
+
y x
(a) e 2x ( cos2x k 2 sin2x ) + (c) ( cos x − sin ) x − y 2 cos x
k
1
1
y x
1
e x (2cos x + sin ) x (d) ( sin x − cos ) x + y 2 cos x
[JAM CA 2008]
+
k
(b) e 2x ( cos2x k 2 sin2x ) +
( ))
1
( ),
W
43. Let x ( y x y x is the
1
2
e x (2cos x − sin ) x Wronskian formed for the solutions
y
y 1 ( ) x and ( ) x of the differential
2
+
k
1
11
(c) e − 2x ( cos2x k 2 sin2x ) − equation y + a y + a y = 0 . If
1
2
1
W  0 for some x = x in [a,b] then
0
e x (2cos x − sin ) x

(a) it vanishes for any x   , a b
+
k
(d) e − 2x ( cos2x k 2 sin2x ) + (b) it does not vanish only at x a
=
1
(c) it does not vanish for any

e x (2cos x + sin ) x x   , a b
=
[JAM CA 2006] (d) it does not vanish only at x b
[JAM CA 2009]
41. The general solution of the
differential equation 44. The general solution of
1
11
y
y 111 + y − y − = 0 is y − 11 m y = 2 0 is
+
c
(a) sinhmx c coshmx
1 2
+
c +
2
x
(a) ( 1 xc + 2 x c 3 )e (b) cosc 1 mx c 2 sinmx
+
(c) cosc 1 mx c 2 sinhmx
−
c +
2
x
(b) ( 1 xc + 2 x c 3 )e
+
(d) sinc 1 mx c 2 coshmx
−
x
(c) c e + x (c + xc )e
2
3
1
[JAM CA 2009]
x
(d) (c + 1 xc 2 )e + x c e
3

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