Page 127 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

256. The number of boundary conditions (d) The solutions are not dependent
required to solve the differential on the boundary conditions.
 2   2 
equation + is [GATE-2016, 2 MARKS]
 x 2  y 2
259. Solution of Laplace’s equation
(a) 2 (b) 0 having continuous second-order
partial derivatives are called
(c) 4 (d) 1

(a) biharmonic functions
[GATE-2001 (CE)]
(b) harmonic functions
257. The solution of the partial differential
 u  2 u (c) conjugate harmonic functions
equation =  is of the form
t   x 2
(d) error functions
 ( k / ) x − ( k / ) x 
( ) C e
(a) cos kt    1 + C e    [GATE-2016; 2 MARKS]
C
2
260. Solution of the differential equation
dy
−
 ( k / ) x − ( k / ) x  = e x y (e − x e y ) is
(b) Ce kt    C e + C e    dx
1
2
x
y e
x
(a) e e = e e x (e − ) 1 + C
(c)
kt  ( ) ( )  y x x e x
−
+
Ce    C 1 cos / k  x C 2 sin − / k  x   (b) e = e − e + C
x
y
(d) (c) e = (e − ) 1 + ce − e x + C
(
)
  (d) ye = e e − 1 +
x
x
ex
e
+
C sin kt  ( ) C 1 cos ( / k  ) x C 2 sin − ( / k  ) x   C


[GATE-2016-CE-SET 1] 261. Solution of the equation
dy 1 1
y
258. Which one of the following is a + tan y = 2 tan sin y is
property of the solutions to the dx x x
Laplace equation:  2 f = 0 ? (a) 2x = sin y (1 2cx+ 2 )
(a) The solutions have neither 2
maxima nor minima anywhere except (b) 2x = sin y (1 cx+ )
at the boundaries.
(c) 2x + sin y (1 cx+ 2 ) 0=
(b) The solutions are not separable in
the coordinates. (d) x + 2sin y (1 2cx+ 2 ) 0=

(c) The solutions are not continuous.

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