M312 #002 Sp17
Partial Differential Equations – Homework #1
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Homework #1 – Review
1. Compute the following integrals.
(a) ∫ 𝑥 3 cos(2𝑥) 𝑑𝑥
𝐿
(b) ∫(𝑥 3 + 𝑥)𝑒 −𝑥 𝑑𝑥
2𝑚𝜋𝑥
2𝑛𝜋𝑥
2. Compute the integral: ∫0 sin ( 𝐿 ) cos (
There are two cases: 𝑚 = 𝑛 and 𝑚 ≠ 𝑛.)
𝐿
𝑚𝜋𝑥
𝑛𝜋𝑥
3. Compute the integral ∫0 sin ( 𝐿 ) sin (
are two cases: 𝑚 = 𝑛 and 𝑚 ≠ 𝑛.)
𝐿
𝑚𝜋𝑥
𝐿
𝐿
) 𝑑𝑥 for any integers 𝑚, 𝑛 ≥ 0. (Hint:
) 𝑑𝑥 for any integers 𝑚, 𝑛 > 0. (Hint: There
𝑛𝜋𝑥
4. Compute the integral ∫0 cos ( 𝐿 ) cos ( 𝐿 ) 𝑑𝑥 for any integers 𝑚, 𝑛 ≥ 0. (Hint: There
are three cases: 𝑚 = 𝑛 = 0, 𝑚 = 𝑛 ≠ 0, and 𝑚 ≠ 𝑛.)
5. Compute the general solution of the following differential equations.
(a) 𝑦 ′ − 3𝑦 = 2𝑒 𝑡
(c) 𝑦 ′ = 𝑥 + 𝑥𝑦 2
2
(b) 𝑦 ′ + 2𝑡𝑦 = 2𝑡𝑒 −𝑡
(d) 𝑦 ′ = 2𝑥𝑦 2
6. Consider the IVP: 𝑦 ′′ − 9𝑦 = 0, 𝑦(0) = 7, 𝑦 ′ (0) = −9.
(a) Solve the IVP using its characteristic equation and exponential functions.
(b) Solve the IVP using its characteristic equation and hyperbolic functions.
(c) Show that the solutions in parts (a) and (b) are equivalent.
(d) Solve the IVP using Laplace transforms.
7. Solve the IVP 𝑦 ′′ − 4𝑦 = 0, 𝑦(3) = 1, 𝑦 ′ (3) = 10 using its characteristic equation and
shifted hyperbolic functions.
8. Compute the general solution of the following Cauchy-Euler equations.
(a) 𝑡 2 𝑦 ′′ − 2𝑦 = 0
(c) 𝑡 2 𝑦 ′′ + 3𝑡𝑦 ′ + 6𝑦 = 0
(b) 𝑡 2 𝑦 ′′ − 3𝑡𝑦 ′ + 4𝑦 = 0
(d) 𝑡 3 𝑦 ′′′ + 𝑡 2 𝑦 ′′ − 2𝑡𝑦 ′ + 2𝑦 = 0
9. Consider the IVP: 𝑢′′ + 𝑢 = cos 𝑡, 𝑢(0) = 𝑢′ (0) = 0.
(a) Solve the IVP using undetermined coefficients.
(b) Solve the IVP using variation of parameters and Cramer’s rule.
10. Choose any method to solve: 𝑢′′ + 5𝑢′ + 6𝑢 = 6 + 2𝑒 −𝑡 , 𝑢(0) = 4, 𝑢′ (0) = −6.
11. A mass weighing 2 lbs. stretches a spring 6 in. If the mass is then pulled down an
additional 3 in. and then released, compute a formula for the position of the mass at any
time 𝑡. *
12. A mass weighing 3 lbs. stretches a spring 3 in. If the mass is pushed upward,
compressing the spring 1 in. and then given an initial downward velocity of 2 ft./s,
compute a formula for the position of the mass at any time 𝑡. *
* Assume there is no damping, 𝑔 = 32 ft./s2, and that the positive direction is downward.