Partial Differential Equations – Homework #1 KEY
M312 #002 Sp17
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Homework #1 KEY – Review
1.
1
3
3
3
(a) 2 𝑥 3 sin(2𝑥) + 4 𝑥 2 cos(2𝑥) − 4 𝑥 sin(2𝑥) − 8 cos(2𝑥) + 𝐶
(b) −𝑥 3 𝑒 −𝑥 − 3𝑥 2 𝑒 −𝑥 − 7𝑥𝑒 −𝑥 − 7𝑒 −𝑥 + 𝐶
2. 0
𝐿
3. ∫0 sin (
4.
𝑚𝜋𝑥
𝐿
) sin (
𝑛𝜋𝑥
𝐿
0, 𝑚 ≠ 𝑛
) 𝑑𝑥 = {
𝐿/2 𝑚 = 𝑛
𝐿
𝑚𝜋𝑥
𝑛𝜋𝑥
∫0 cos ( 𝐿 ) cos ( 𝐿 ) 𝑑𝑥
𝐿,
𝑚=𝑛=0
= {𝐿/2, 𝑚 = 𝑛 ≠ 0
0,
𝑚≠𝑛
5.
1
(a) 𝑦 = 𝐶𝑒 3𝑡 − 𝑒 𝑡
2
(b) 𝑦 = 𝑡 2 𝑒 −𝑡 + 𝐶𝑒 −𝑡
2
(c) 𝑦 = tan (2 𝑥 2 + 𝐶)
1
(d) 𝑦 = 𝐶−𝑥 2 , 0
6.
(a) 𝑦 = 2𝑒 3𝑡 + 5𝑒 −3𝑡
(b) 𝑦 = 7 cosh(3𝑡) − 3 sinh(3𝑡)
(c) Good luck.
(d) Refer to part (a) or (b).
7. 𝑦 = cosh(2(𝑡 − 3)) + 5 sinh(2(𝑡 − 3))
8.
(a) 𝑦 = 𝑐1 𝑡 2 + 𝑐2 𝑡 −1
(b) 𝑦 = 𝑐1 𝑡 2 + 𝑐2 𝑡 2 ln 𝑡
(c) 𝑦 = 𝑐1 𝑡 −1 cos(√5 ln 𝑡) + 𝑐2 𝑡 −1 sin(√5 ln 𝑡)
(d) 𝑦 = 𝑐1 𝑡 + 𝑐2 𝑡 2 + 𝑐3 𝑡 −1
1
9. 𝑢 = 2 𝑡 sin 𝑡
10. 𝑢 = 1 + 𝑒 −𝑡 + 𝑒 −2𝑡 + 𝑒 −3𝑡
1
11. 𝑦 = 4 cos(8𝑡) ft.
1
1
12. 𝑦 = − 12 cos(8√2𝑡) + 4√2 sin(8√2𝑡) ft.