Unit VI: Applications of Partial Differential Equations
Multiple Choice Questions
1) The partial differential equation of one-dimensional wave is. (01)
u 2 2u
a) =c
t x2
2u
2 u
2
b) = c
t2 x2
u u
c) + =0
t x
2u 2 u
d) 2 + c x = 0
x
2y
2) The most suitable general solution of one-dimensional wave equation, = c2
t2
2y
, where y(x, t) be the deflection of string, which satisfies all boundary and
x2
initial conditions, is (01)
a) y(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct )
b) y(x, t) = C1 cos mx + C2 sin mx + C3 cos mct + C4 sin mct
2 2
c) u (x, t) = ( C1 cos mx + C2 sin mx ) e– c m t
d) u(x, y) = ( C1 cos mx + C2 sin mx ) ( C3 cosh my + C4 sinh my )
3) The partial differential equation of one-dimensional heat is (01)
u 2 2u
a) =c
t x2
2u
2 u
2
b) = c
t2 x2
u u
c) + =0
t x
2u 2 u
d) 2 + c x = 0
x
4) If u ( x , y ) = 0 as y → , x in (0,L) then the most suitable general solution of two
2u 2u
dimensional heat equation , + = 0 , where u(x, y) be the
x2 y2
the temperature of plate, which satisfies all boundary and initial conditions, is. (01)
a) u(x, y) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct )
b) u (x , y) = ( C1 emx + C2 e–mx ) ( C3 cos my + C4 sin my )
2 2
c) u (x , t) = ( C1 cos mx + C2 sin mx) e– c m t
d) u(x , y) = ( C1 cos mx + C2 sin mx ) ( C3 emy + C4 e – my )
5) The most suitable general solution of one-dimensional heat equation, (01)
u u
2
= C2 , where u(x, t) be the temperature of a rod, which satisfies all
t x2
boundary and initial conditions is.
a) u(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct )
b) u(x, y) = C1 cos mx + C2 sin mx + C3 cos mct + C4 sin mct
2 2
c) u (x, t) = ( C1 cos mx + C2 sin mx ) e– c m t
d) u(x, y) = ( C1 cos mx + C2 sin mx ) ( C3 cosh my + C4 sinh my )
2y 2y
6) The most suitable general solution of one-dimensional wave equation, =9 2 ,
t2 x
where y(x, t) be the deflection of string, which satisfies all boundary and initial
conditions, is (01)
a) y(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos 3mt + C4 sin 3mt )
b) y(x, t) = C1 cos mx + C2 sin mx + C3 cos 3mt + C4 sin 3mt
– 9 m2 t
c) y(x, t) = ( C1 cos mx + C2 sin mx ) e
d) y(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos 9mt + C4 sin 9mt )
7) If u ( x , y ) = 0 as x → , x in (0,L) then the most suitable general solution of two-
2u 2u
dimensional heat equation , + =0 ,
x2 y2
where u (x, y) be the temperature of plate, which satisfies all boundary and initial
conditions, is. (01)
a) u(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct )
b) u (x , y) = ( C1 emx + C2 e–mx ) ( C3 cos my + C4 sin my )
2 2
c) u (x , t) = ( C1 cos mx + C2 sin mx) e– c m t
d) u(x , y) = ( C1 cos mx + C2 sin mx ) ( C3 emy + C4 e – my )
2y 2
2y
8) If the most suitable general solution of one-dimensional wave Equation, =c
t2 x2
which satisfies all boundary and initial conditions, is.
y(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct ), then by using
boundary condition y(0 , t) = 0 ; t , the value of arbitrary constant C1 is ----- (01)
a) − 𝑛𝜋⁄𝐿
b) 𝑛𝜋
c) 𝑛𝜋⁄
𝐿
d) 0
9) The most suitable general solution of one-dimensional heat equation,
u 2u
= 25 2 , where u(x, t) be the temperature of a rod, which satisfies all boundary
t x
and initial conditions is. (01)
a) u(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos 5mt + C4 sin 5mt )
b) u(x, y) = C1 cos mx + C2 sin mx + C3 cos 5mt + C4 sin 5mt
2
c) u (x, t) = ( C1 cos mx + C2 sin mx ) e– 25m t
2
d) u (x, t) = ( C1 cos mx + C2 sin mx ) e– 5 m t
10) If the most suitable solution of one dimensional heat equation : (01)
u 2u
= c2 , where u(x, t) be the temperature of a rod, which satisfies all boundary and
t x2
2 2
initial conditions is. u (x , t) = ( C1 cos mx + C2 sin mx) e– c m t
Then by using boundary condition u(0, t) = 0; t , the value of arbitrary constant C1 is –
a) − 𝑛𝜋⁄𝐿
b) 𝑛𝜋
c) 𝑛𝜋⁄
𝐿
d) 0
2y 2 y
2
11) The differential equation of stretched string is , t2 = c x2 (01)
with the boundary conditions ,(i)y(0, t) = a sin pt ; t (ii) y(l, t) = a sin pt ; t .
where , y(x, t) - be the displacement of string at any time t.
The most suitable solution is
y(x, t) = (C1 cos mx + C2 sin mx) (C3 cos mct + C4 sin cmt) …(1)
By using condition ( i ), we get
a sin pt = C1 C3 cos cmt + C1 C4 sin cmt. Then the value of C3 is equal to
a) Cm
b) 0
a
c) C
1
d) None of the above
12) If the most suitable general solution of two dimensional heat equation : (01)
u
2
u2
2 + = 0 where u(x, t) be the temperature of a plate, which satisfies all
x y2
Boundary and initial conditions and as y → , solution is
u(x, y) = ( C1 cos mx + C2 sin mx ) (C3 emy + C4 e– my ) …(1)
Then by using boundary condition u (x, ) = 0; x , the value of arbitrary constant
C1 is -----
a) − 𝑛𝜋⁄𝐿
b) 𝑛𝜋
c) 𝑛𝜋⁄
𝐿
d) 0
13) If the most suitable solution of one dimensional heat equation : (01)
u u2
= c2 , where u(x, t) be the temperature of a rod, which satisfies all boundary
t x2
2 2
and initial conditions is. u (x , t) = ( C1 cos mx + C2 sin mx ) e– c m t ……… (1)
By using conditions.
(i) u (0, t) = 0 ; t (ii) u (L, t) = 0 ; t we get C1 = 0 and m = n ; n = 1, 2, 3, . . .
Then equation (1) becomes
2
( n ) c2 t
(a) u (x, t) = bn sin (n) x e
n=1
– ( n ) 2 c2 t
(b) u (x, t) = bn sin (n ) x e
n=1
– ( n ) 2 c2 t
(c) u (x, t) = bn cos (n ) x e
n=1
(d) u (x, t) = bn sin (n ) x
n=1
2u 2u
14) For the equation x2 + y2 = 0 with condition (01)
(i)u = 0 when y→ ∞ ∀ 𝑥 then………
(a) C3 = 0
(b) C2 = 0
(c) C1 = 0
(d) C4 = 0
15) An infinitely long uniform metal plate is enclosed between lines y = 0 and y = L for
x > 0. The temperature is zero along the edges: y = 0, y = L and at infinity. The most
suitable general solution is (01)
a) u(x, t) = ( C1 cos mx + C2 sin mx ) ( C3 cos mct + C4 sin mct )
b) u (x , y) = ( C1 emx + C2 e–mx ) ( C3 cos my + C4 sin my)
2 2
c) u (x , t) = ( C1 cos mx + C2 sin mx) e– c m t
d) u(x , y) = ( C1 cos mx + C2 sin mx ) ( C3 emy + C4 e – my )
𝑛𝜋𝑥 𝑛𝜋𝑥
16) If u(x, t) = bn sin 𝑙
cos 𝑙 then by using condition (01)
n=1
𝑛𝜋𝑥
u(x, 0) = a sin , u(x, t) = ………………….
𝑙
𝜋𝑥 𝜋𝑐𝑡
a) u(x, t) = a sin 𝑙 cos 𝑙
𝜋𝑥
b) u(x, t) = sin 𝑙
𝜋𝑐𝑡
c) u(x, t) = a sin cos 𝑙
1 𝜋𝑐𝑡
d) u(x, t) = a sin 𝑙
cos 𝑙
2u 2u
17) 2 + = 0 two dimensional heat flow is also known as (01)
x y2
a) Fourier equation
b) Laplace equation
c)Wave equation
d) None of these
18) An infinitely long uniform metal plate is enclosed between lines y = 0 and y= l for x > 0. The
temperature is 0 along the edges y= 0, y= l and at infinity. If the edge x= 0 is kept at a constant
temperature u0 then using the most general solution u(x, y) = (C1 𝑒 𝑚𝑥 + C2 𝑒 − 𝑚𝑥 ) (C3 cos my+ C4 sin
my) the value of C3=……. (01)
a) 0
b) 1
c) 2
d) 3
2y 2 y
2
19) The differential equation of stretched string is , t2 = c x2 (01)
with the boundary conditions , (i) y(0, t) = 0 t (ii) y(l, t) = 0 ; t .
where , y(x, t) - be the vibration of string of length l fixed t both ends
By using condition (i) we get, The most suitable solution is
y(x, t) = (C1 cos mx + C2 sin mx) (C3 cos mct + C4 sin cmt) …(1)
we get C1 = 0 then by using (ii)C4 =………………..
a) 0 b) 1 c) – 1 d) 2
u 2u
20) For t = c2 2 if (i) u is finite ∀ t, (ii) u(0,t) = 0 ∀ t, (iii) u(𝜋, 𝑡)= 0 ∀ t, (01)
x
– m2 t
The most general solution is, u(x ,t) = ( C1 cos mx + C2 sin mx) e
By using (i) & (ii) the solution becomes,
– m2 t
u(x ,t)= C2 sin mx e
by using (iii) , the value of m = …………
a) N𝜋
b) 0
c) n
d) 1
Answers:
1-b 2-a 3-a 4-d 5-c 6-a 7-b 8-d 9-c 10-d
11-b 12-d 13-b 14-a 15-b 16-a 17-b 18-a 19-a 20-c
