Department of Mathematics, IIT Madras
MA2020, Differential Equations
Assignment-6
Exercises for Sections 6.1-6.3
1. Solve the problem for a semi-infinite string with free end:
𝑢𝑡𝑡 = 𝑐 2𝑢𝑥𝑥 for 𝑥 > 0, 𝑡 > 0; 𝑢𝑥 (0, 𝑡) = 0 for 𝑡 ≥ 0; 𝑢 (𝑥, 0) = 𝑓 (𝑥) for 𝑥 > 0; and
𝑢𝑡 (𝑥, 0) = 𝑔(𝑥) for 𝑥 > 0. ∫ 𝑥+𝑐𝑡
Ans: 𝑢 (𝑥, 𝑡) = 21 𝑓 (𝑥 − 𝑐𝑡) + 𝑓 (𝑥 + 𝑐𝑡) + 2𝑐1 𝑥−𝑐𝑡 𝑔(𝑠) 𝑑𝑠 for 𝑥 ≥ 𝑐𝑡;
∫ 𝑐𝑡−𝑥 ∫ 𝑐𝑡+𝑥
𝑢 (𝑥, 𝑡) = 21 𝑓 (𝑐𝑡 − 𝑥) + 𝑓 (𝑐𝑡 + 𝑥) + 2𝑐1 0 𝑔(𝑠) 𝑑𝑠 + 0 𝑔(𝑠) 𝑑𝑠 for 0 < 𝑥 < 𝑐𝑡.
2. Solve the following problem for a semi-infinite string with a fixed end:
𝑢𝑡𝑡 = 𝑐 2𝑢𝑥𝑥 for 𝑥 > 0, 𝑡 > 0; 𝑢 (0, 𝑡) = 0 for 𝑡 ≥ 0; 𝑢 (𝑥, 0) = 𝑓 (𝑥) for 𝑥 > 0; and
𝑢𝑡 (𝑥, 0) = 𝑔(𝑥) for 𝑥 > 0. ∫ 𝑥+𝑐𝑡
Ans: 𝑢 (𝑥, 𝑡) = 12 𝑓 (𝑥 − 𝑐𝑡) + 𝑓 (𝑥 + 𝑐𝑡) + 2𝑐1 𝑥−𝑐𝑡 𝑔(𝑠) 𝑑𝑠 for 𝑥 ≥ 𝑐𝑡;
∫ 𝑐𝑡+𝑥
𝑢 (𝑥, 𝑡) = 21 𝑓 (𝑥 + 𝑐𝑡) − 𝑓 (𝑐𝑡 − 𝑥) + 2𝑐1 𝑐𝑡−𝑥 𝑔(𝑠) 𝑑𝑠 for 0 < 𝑥 < 𝑐𝑡.
3. Use the method of separation of variables to solve the following PDEs:
(a) 𝑦 3𝑢𝑥 + 𝑥 2𝑢𝑦 = 0, 𝑢 (𝑥, 0) = exp(𝑥 3 /3). Ans:𝑢 (𝑥, 𝑦) = exp(𝑥 3 /3 − 𝑦 4 /4).
(b) 𝑢𝑥 = 2𝑢𝑡 + 𝑢, 𝑢 (𝑥, 0) = 6 exp(−3𝑥). Ans: 𝑢 (𝑥, 𝑡) = 6 exp(−3𝑥 − 2𝑡).
(c) 𝑢𝑥 + 3𝑢𝑦 = 0, 𝑢 (0, 𝑦) = 4 exp(−2𝑦) + 3 exp(−6𝑦).
Ans: 𝑢 (𝑥, 𝑦) = 4 exp(6𝑥 − 2𝑦) + 3 exp(18𝑥 − 3𝑦).
(d) 4𝑢𝑥 + 𝑢𝑦 = 3𝑢, 𝑢 (0, 𝑦) = 3 exp(−𝑦) − exp(−5𝑦).
Ans: 𝑢 (𝑥, 𝑦) = 3 exp(𝑥 − 𝑦) − exp(2𝑥 − 5𝑦).
(e) 𝑢𝑥𝑥 = 𝑢𝑦 + 2𝑢, 𝑢𝑥 (0, 𝑦) = 1 + exp(−3𝑦), 𝑢 (0, 𝑦) = 0.
√ √
Ans: 𝑢 (𝑥, 𝑦) = 8−1/2 exp( 2 𝑥) − exp(− 2 𝑥) + exp(−3𝑦) sin 𝑥.
4. Show that if the method of separation of variables is used on the PDE
𝑢𝑥 + (𝑥 + 𝑦)𝑢𝑦 = 0, then one gets the only solution as 𝑢 = 𝑐, a constant.
5. A tightly stretched string of length ℓ has its ends fixed at 𝑥 = 0 and 𝑥 = ℓ. The mid-point
of the string is then taken to a height ℎ and then released with initial velocity 0. Find the
displacement of the string at each 𝑥 and each 𝑡.
Í∞
Ans: 𝑢 (𝑥, 𝑡) = 𝑛=1 𝑐𝑛 sin(𝑛𝜋𝑥/ℓ) cos(𝑛𝜋𝑐𝑡/ℓ), 𝑐𝑛 = (8ℎ/𝑛 2 𝜋 2 ) sin(𝑛𝜋/2).
6. If a string of length ℓ is initially in the equilibrium position and each of its points is given
the velocity sin3 (𝜋𝑥/ℓ). Find the displacement of the string at each 𝑥 and each 𝑡.
3ℓ 𝜋𝑥 𝜋𝑐𝑡 ℓ 3𝜋𝑥 3𝜋𝑐𝑡
Ans: 𝑢 (𝑥, 𝑡) = sin sin − sin sin .
4𝜋𝑐 ℓ ℓ 12𝜋𝑐 ℓ ℓ
7. A uniform string of length ℓ, constant density 𝜌 and tension 𝑇0 has its equilibrium position
along the 𝑥-axis with fixed end-point 𝑥 = 0 and free end-point 𝑥 = ℓ. At 𝑥 = ℓ, the
string is 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, then find the subsequent motion of the string.
∞
∑︁ (2𝑛 − 1)𝜋𝑥 (2𝑛 − 1)𝜋𝑐𝑡 𝑇0
Ans: 𝑢 (𝑥, 𝑡) = 𝑎𝑛 sin cos , where 𝑐 2 = and
2ℓ 2ℓ 𝜌
𝑛=1
32 ℎℓ 3 6(−1)𝑛 + (4𝑛 − 2)𝜋
𝑎𝑛 = for 𝑛 = 1, 2, 3, . . ..
𝜋 4 (2𝑛 − 1) 4
1
8. A tightly stretched string with fixed end-points 𝑥 = 0 and 𝑥 = 1 is initially in a position
given by 𝑓 (𝑥) = 𝑥 for 0 ≤ 𝑥 ≤ 1/2, and 𝑓 (𝑥) = 1 − 𝑥 for 1/2 < 𝑥 ≤ 1. If it is released
from this position with velocity 𝑎, a constant, perpendicular to the 𝑥-axis, then find the
displacement at each 𝑥 and each 𝑡.
∑︁∞
Ans: 𝑢 (𝑥, 𝑡) = sin(𝑛𝜋𝑥) 𝑎𝑛 cos(𝑛𝜋𝑐𝑡) + 𝑏𝑛 sin(𝑛𝜋𝑐𝑡) , for 𝑛 = 1, 2, 3, . . .,
𝑛=1
𝑎𝑛 = 4/(𝑛 2 𝜋 2 ) sin(𝑛𝜋/2) and 𝑏𝑛 = 2𝑎/(𝑛 2 𝜋 2𝑐) 1 + (−1)𝑛+1 .
