MA2020-Differential Equations

Assignment 8

1. Use D’Alembert’s formula to solve

(a)

utt = a2 uxx , −∞ < x < ∞, t > 0

u(x, 0) = x, ut (x, 0) = 4ax, −∞ < x < ∞

(b)

utt = a2 uxx , −∞ < x < ∞, t > 0

u(x, 0) = sin(x), ut (x, 0) = a cos(x), −∞ < x < ∞

2. Use the method of separation of variables to solve the following problems:

(a) y 3 zx + x2 zy = 0, z(x, 0) = exp (x3 /3)

(b) ux = 2ut + u, u(x, 0) = 6 exp (−3x)

(c) ux + 3uy = 0, u(0, y) = 4 exp(−2y) + 3 exp(−6y)
(d) 4ux + uy = 3u, u(0, y) = 3 exp(−y) − exp(−5y)

(e) uxx = uy + 2u, ux (0, y) = 1 + exp(−3y), u(0, y) = 0

Ans:
x3 y4
(a) z(x, y) = e 3 − 4

(b) u(x, t) = 6e−3x−2t

(c) u(x, y) = 4e6x−2y + 3e18x−3y
(d) u(x, y) = 3ex−y − e2x−5y
h √ √ i
(e) u(x, y) = 1
√
2 2
e 2x
− e− 2x
+ e−3y sin x
3. A tightly stretched string of length l has its ends fastened at x = 0 and x = l. The mid
point of the string is then taken to height h and then released with initial velocity zero. Find

the displacement of the string at every x and at every time t.

Ans:
∞
nπx nπct 8h nπ
X      
u(x, t) = cn sin cos , cn = 2 2 sin
n=1 l l nπ 2
4. If a string of length l is initially at rest in the equilibrium position and each of its points

is given the velocity sin3 (πx/l) . Find the displacement of the string at every x and every t.
Ans:
! !
3l πx πct l 3πx 3πct
       
u(x, t) = sin sin − sin sin
4πc l l 12πc l l
5. A uniform string of length l, constant density ρ and tension T0 has its equilibrium position
along the x-axis, with end at x = 0 fixed and the end at x = l free to move (in a plane)

perpendicular to the x-axis in such a way that the end of the string remains parallel to the
x-axis. If u(x, t) is the transverse displacement of the string at a point x and time t > 0,
and released from rest while in the displaced position u(x, 0) = hx(l − x)2 , where h > 0 is

small, find the subsequent motion of the string.

Ans:
∞
! !
(2n − 1)πx (2n − 1)πct
, c2 = T0 /ρ,
X
u(x, t) = an sin cos
n=1 2l 2l
32hl3 6(−1)n + (4n − 2)π
" #
an = , n = 1, 2, 3 · · ·
π4 (2n − 1)4

6. A tightly stretched string with fixed end points x = 0 and x = 1 initially in a position
given by
1


 x, 0 ≤ x ≤ 2
f (x) =
1
1 − x, ≤ x ≤ 1.


2
If it is released from this position with velocity a, a constant, perpendicular to x-axis, find

the displacement at every x and every time t.

P∞
Ans: u(x, t) = n=1 sin(nπx)[an cos(nπct) + bn sin(nπct)],
     
4 nπ 2a
where an = n2 π 2
sin 2
, bn = n2 π 2 c
[1 + (−1)n+1 ] , n = 1, 2, 3 · · ·

7. Solve

ut = c2 uxx , 0 < x < l, t > 0

ux (0, t) = 0, t ≥ 0

u(l, t) = 0, t ≥ 0
 u(x, 0) = x, 0 < x < l

Ans:
∞
c2 (2n − 1)2 π 2 t
" # " #
X (2n − 1)πx
u(x, t) = An cos exp − ,
n=1 2l 4l2
8l 4l(−1)n
An = − − , n = 1, 2, 3, · · ·
(2n − 1)2 π 2 (2n − 1)π

8. A homogeneous rod of conducting material of length 100cm has its ends kept at zero

temperature and the temperature initially is



 x, 0 ≤ x ≤ 50
f (x) =
100 − x, 50 ≤ x ≤ 100.



Find the temperature u(x, t) at every x and every time t.

Ans:
∞
n2 π 2 c2 t
!
nπx
X  
u(x, t) = cn sin exp − ,
n=1 l l2
4l nπ
 
cn = 2 2 sin
nπ 2

9. A metal bar of length l has its ends x = 0 and x = l insulated. Initially the bar is at
 
3πx
temperature given by 6+4 cos l
, 0 < x < l. Assuming the surface of the bar is insulated,
find the temperature everywhere in the bar at time t.

Ans:
n 2 π 2 c2 t
!
3πx
 
u(x, t) = 6 + 4 cos exp −
l l2
10. Solve

ut = uxx , 0 < x < 10, t > 0

u(0, t) = 90, t ≥ 0

u(10, t) = 60, t ≥ 0

u(x, 0) = 5x + 50, 0 < x < 10

Ans:
∞
n2 π 2 c2 t
!
nπx 80
 
[(−1)n+1 − 1], n = 1, 2, 3, · · ·
X
u(x, t) = −3x + 90 + cn exp − sin , cn =
n=1 100 10 nπ

11. The ends A and B of a rod [20 cm long] are maintained at 300 c and 800 c, respectively
until steady state prevails. The temperatures of the ends are then suddenly changed to 400 c

and 600 c, respectively. Find the temperature distribution in the rod at time t.
Ans
∞
n 2 π 2 c2 t
!
nπx 20
 
[2(−1)n+1 − 1], n = 1, 2, 3, · · ·
X
u(x, t) = x + 40 + cn sin exp − 2
, cn =
n=1 l l nπ

12. The temperature at one end of a bar of 50cm long is kept at 00 c and other end is kept
at 1000 c until steady state conditions prevail. The two ends of the bar are then suddenly

insulated. Find the temperature distribution in the rod at time t.

Ans: l = 50 and
∞
n 2 π 2 c2 t
!
nπx 4l
X  
u(x, t) = An cos exp − 2
, A0 = l, An = 2 2 (cos nπ − 1), n = 1, 2, 3, · · ·
n=0 l l nπ