Page 129 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations
: →
269. The equation of the curve satisfying 272. Let y R R be a solution of the
the differential equation d y
2

=
−
x
QDE, − y e , x R ,
 dy  2 dy dx 2
y       + (x − ) y − x = 0 can be y ( ) 0 = y 1 ( ) 0 = 0 . Then which of the
 dx  dx
a following are true?
(a) y attains its minimum on R.
(a) Circle (b) Straight line
(b) y is bounded on R.
(c) Parabola (d) Ellipse 1
( )
−
x
(c) lime y x = .
270. For the boundary value problem, x→  4
y + 11  y = 0 , (d) lime y x = 1
( )
x
)
)
( ), y −
1
( y  = y  1 (  = y  x→− 4
−
( ) .
273. Consider the Lagrange equation
To each eigen value  , there z  z 
corresponds x 2 + y 2 = (x + ) y z . Then the
 x  y
(a) only one eigen function general solution of the given equation
is
(b) two eigen functions  xy x − y 
(a) F   ,     = 0 for an
(c) two linearly independent eigen  z z 
functions arbitrary differentiable function F.

(d) two orthogonal eigen functions   x − y 1 1  
(b) F   , −   = 0 for an
2
d y  z x y 

271. Let − q ( ) x y = 0, 0 x , arbitrary differentiable function F.
dx 2
dy   1 1  
y ( ) 0 = 1, ( ) 0 = 1, where q(x) is (c) z = f   −   for an arbitrary
dx  x y 

monotonically increasing continuous differentiable function f.
function. Then,  1 1 
(d) z = xy f    −    for an
y
(a) ( ) x →  as x →   x y 
arbitrary differentiable function f.
dy
(b) →  as x → 
dx 274. Which of the following are complete
integral of the partial differential
(c) y(x) has finitely many zeros in 2
)
0, equation pqx + yq = 1.

(d) y(x) has infinitely many zeros in
)
0, .

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