Page 132 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations

286. Which one of the following is a 289. Consider the following partial
property of the solutions to the differential equation u(x, y) with the
Laplace equation:  2 f = 0 ? constant c > 1:

 u  u
(a) The solutions have neither  y + c  x = 0

maxima nor minima anywhere except
at the boundaries. Solution of this equation is

(b) The solutions are not separable in (a) ( ,u x y = ) ( f x cy+ )
the coordinates.
)
−
u
(b) ( , x y = ) ( f x cy
(c) The solutions are not continuous.
u
(c) ( , x y = ) ( f cx + ) y
(d) The solutions are not dependent
on the boundary conditions. (d) ( , x y = ) ( f cx − ) y
u

[GATE – 2016, 2 MARKS]
[GATE – 2017 – ME – SESSION-1]
287. Solution of Laplace’s equation 290. The complete integral of
having continuous second-order 3 2
−
2
partial derivatives are called (z − px qy ) = pq + ( 2 p + ) q is
(a) biharmonic functions 2
+
2
(a) z = ax by + 3 pq + ( 2 p + ) q
(b) harmonic functions

+
(c) conjugate harmonic functions (b) z = ax by + 3 ab + ( 2 a + ) b 2
2
(d) error functions
(c) z = ax by + 3 ab + 3 ( 2 a + ) b 2
+
2
[GATE – 2016; 2 MARKS]
=
+
+
288. Consider the following partial (d) z ax by c
differential equation
[ESE – 2017 (Common Paper)]
 2   2   2 
0
3 + B + 3 + 4 = 291. The solution of the following partial

 x 2  x y  y 2  2 u  2 u
differential equation 2 = 9 2 is
For the equation to be classified as  x  y
2
parabolic, the value of B must be (a) sin (3x − (b) 3x + 2
2
____________. ) y y

2
[GATE – 2017 – CE – SESSION-1] (c) sin (3x − 3y ) (d) (3y − x 2 )

[ESE – 2017 (Common Paper)]

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