Page 122 - Engineering Mathematics Workbook_Final

Differential Equations & Partial Differential Equations
)
(d) ( ) x = n K exp − ( x L dy
/
boundary conditions = 1 and
dx x=
[GATE-2010-EC] 0
dy
228. If the characteristic equation of the dx x=  = − 1 has

differential equation 2
2
d y + 2 dy + y = 0 has two equal (a) no solution
dx 2 dx
roots, then the values of  are (b) exactly two solutions

(a) 1 (b) 0, 0 (c) exactly one solution



(c) j (d) 1/ 2 (d) infinitely many solutions
[GATE 2017]
[GATE-2014-EC-SET 2]
−
−
x
x
231. If e and xe are two independent
229. Consider two solution ( ) t = x 1 ( ) t d y dy
x
2
and ( ) t = x 2 ( ) t of the differential solutions of dx 2 +  dx + y = 0 then
x
2
d x ( ) t the value of  = ____ .
( ) 0, t  ,
equation + x t = 0
dt 2 PARTICULAR INTEGRAL
dx ( ) t
such that ( ) 0 = 1, 1 = 0, 232. For initial value problem
x
1
dt ) ) x
t= 0 y + 2y + (101 y = (104 e , y(0) =
dx ( ) t 1.1 and y(0) = -0.9. Various solutions
x = 0, 2 = 1. The are written in the following groups.
2
dt
t= 0 Match the type of solution with the
x 1 ( ) t x 2 ( ) t correct expression.
Wronskian ( ) t = dx 1 ( ) t dx 2 ( ) t Group-1
W
dt dt P. General solution of homogeneous
at t  / 2 is equations
=

(a) 1 (b) -1 Q. Particular integral

(c) 0 (d) / 2 S. Total solution satisfying boundary
conditions
[GATE-2014-ME-SET 3]
Group-II
230. The differential equation x
2
d y + 16y = 0 for y(x) with the two (1) 0.1e
+
dx 2 (2) e − x  cos 10A x B sin10x 

120