Page 117 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

x
e x −
(a) ( 2 y 2 ) = c 203. The differential equation
2
( xy 3 + y cos ) x dx + ( x y +  sin ) x dy = 0
2
(b) ( 2 y 2 ) = c is exact for
e x +
x
−
x
e
(c) ( x + 2 y 2 ) = c (a)  3 ,  = = 1
2
(d) ( x − 2 y 2 ) = c (b)  1,  = = 3
−
x
e
2
200. The solution of 2
( y xy dx − ( x x y dy = 0 is (c)  3 ,  = = 1
)
)
2
+
2
−
 x  x (d)  1,  = = 2
(a) ln       − = c 3 [GATE]
 y  y
204. For the differential equation,
 x  x
(b) ln       + = c f ( , x y ) dy + g ( , x y = ) 0 to be
 y  y dx
exact,
 x 
=
(c) ln       + xy c f   g f   g
 y  (a) = (b) =
 y  x  x  y
 x 
=
(d) ln       − xy c  2 f  2 g
 y  (c) f = g (d)  x 2 =  y 2

EXACT DIFFERENTIAL EQUATION
[GATE-1997-CE]

201. The differential equation
VARIABLE SEPARABLE
( 27x + ky cos ) x dx + ( 2sin x − 27y 3 ) dy = 0
2
is exact for k = ____. 205. The solution of dy = y with initial
2
dx
[GATE]
1
y
value ( ) 0 = bounded in the
202. If the integrating factor of interval
)
( x y + 3y dx + ( 3x y x dy = (a) −   (b) −  1
)
7
2
−
8
0
x
x


is x y then  = ____ and (c) x  1, x  1 (d) 2 x−  
2
 = ____ .
[GATE-2007-ME]
[GATE]

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