Page 121 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

[GATE-1993 (ME)] The form of non-zero solutions y
(where m varies over all integers) are
222. The Solution to the differential

equation (a) y =  A sin m x
x +
x +
( ) 0
f 11 ( ) 4 f 1 ( ) 4 f x = is m m a

f
(a) ( ) x = e − 2x (b) y =  A m cos m x
1
(b) ( ) x = f 1 e 2x , f 2 ( ) x = xe − 2x m a

(c) ( ) x = f 1 e − 2x , f 2 ( ) x = xe − 2x (c) y =  A x m
m
a
m
−
x
(d) ( ) x = f 1 e 2x , f 2 ( ) x = xe

(d) y =  A e m x
[GATE-1995-ME] m m a
223. The complete solution of the ordinary [GATE-2006-EC]
differential equation
2
d y + p dy + qy = 0 is 226. The homogeneous part of the
differential equation
dx 2 dx d y dy
2
y c e + 1 − x c e − 3x . Then, p and q are dx 2 + p dx + qy = (p, q, r are
=
r
2
(a) p = 3, q = 3 (b) p = 3, q = 4 constants) has real distinct roots if
(c) p = 4, q = 3 (d) p = 4, q = 4 2 2
(a) p  4q  0 (b) p  4q  0
[GATE-2005-ME]
=
(c) p  2 4q = 0 (d) p  2 4q r
224. For the equation
t +
1
x 11 ( ) 3x t + ( ) 5 [GATE-2009 (PI)]
( ) 2x t = , the
solution x(t) approaches to the 227. A function n(x) satisfies the
following value as t →  differential equation
2
(a) 0 (b) 5/2 d n ( ) x − n ( ) x = 0 where L is a
dx 2 L 2
(c) 5 (d) 10
constant. The boundary conditions
[GATE-2005-EE] are: n(0) = K and ( ) 0n  = . The

225. For the differential equation solution to this equation is
2
d y + k y = 2 0 the boundary (a) ( ) x = K exp ( /x L
)
n
dx 2
)
conditions are
(b) ( ) x = n K exp − ( / x L
(i) y = 0 for x = 0 and

(ii) y = 0 for x = a (c) ( ) x = n K 2 exp ( x L− / )

119