Page 100 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

2
d y 93. An integrating factor of
89. If + y = 0 under the conditions
dt 2 dy + 2x
dy x (3x + ) 1 y = xe is ________
y = 1, = 0, when t = then y is dx
0
dt
equal to (a) xe (b) 3xe
x
3x
(a) sin t (b) cos t
x
3 x
(c) xe (d) x e
(c) tan t (d) cot t
[ESE 2018 (EE)] [JAM 2005]

90. The solution (up to three decimal 94. The general solution of −
2 11
place) at x = 1 of the differential 5 + 9 = 0 is __________
1
2
d y dy
equation + 2 + y + 0 subject to 3x
dx 2 dx (a) (c + c 2 ) x e
1
boundary conditions ( ) 0 = 1 and
y
3
(b) (c + c ln ) x x
dy ( ) 0 = − 1 is ___________. 1 2
dx
3
(c) (c + c 2 ) x x
1
[GATE-2018 (CE-Morning Session)]
3

91. Given the ordinary differential (d) ( + ) [JAM 2005]
2
1
equation
2
d y x )
+
y
1
2
d y dy 95. 2 − y = x (sin x e , ( ) 0 = ,
+ − 6y = dx
0
dx 2 dx
y 1 ( ) 0 = 1 [JAM 2005]
dy
with ( ) 0 = and ( ) 0 = , the
1
0
y
dx 96. Solve
value of y(1) is _____________ (2 sin x + y 3y 4 sin cos ) x dx −
x
(Correct to two decimal places).
(4y 3 cos x + 2 cos ) x dy = 0 .
[GATE-2018 (ME-Afternoon Session)]
92. The position of a particle y(t) is [JAM 2005]
described by the differential equation: (c + c ln ) x
97. If 1 2 is the general
2
d y dy 5y x
= − − .
dt 2 dt 4 solution of the differential equation
2
d y dy
2
0
The initial conditions are ( ) 0 = 1 x + kx + y = 0, x  then
y
dx 2 dx
dy k equals ______
and = 0 . The position (accurate
dt t= 0
to two decimal places) of the particle [JAM 2006]
at t = is _________.
[GATE-2018 (EC)]

98