Final Exam December 20, 2011

Math 420 - Ordinary Differential Equations

• No credit will be given for answers without mathematical or logical justification.

• Simplify answers as much as possible.

• Leave solutions in implicit form when appropriate.

• Scientific calculators only. No notes or books

Part I (35 points)

Problem 1. (5 pts) Find the real part of eln(7)−π4 i.

Problem 2. (5 pts) Compute, directly from the definitions, the Laplace transform of u2(t).

1

Problem 3. (10 pts) Find the particular solution.

dx

dt=− 2 t x

t2 + 1+ 1, x(0) = 0. (1)

Problem 4. (10 pts) Find the general solution.

dx

dt=

x2 + 2

2 t x(2)

2

Problem 5. (5 pts) Write the following nonlinear second order equation as a system of firstorder equations.

d2y

dt2+

(dy

dt

)2

− y = 0 (3)

3

Part II (65 points)

Problem 6. (65 points total) Consider the damped harmonic oscillator

d2y

dt2+ 2p

dy

dt+ ω2

0y = cos(ωt)u(t) (4)

If ω20 − p2 > 0, meaning the system is underdamped, define ω1 =

√ω20 − p2.

Let z be the fundamental solution of (4). That is, z(t) satisfies d2zdt2 + 2pdz

dt + ω20z = δ(t)

and z(t) = 0 for all t < 0.

6a) (5 pts) Let Z(s) = L(z(t))(s) be the Laplace transform of z(t). Assume the system isunderdamped, and prove that Z(s) = 1

(s+ p)2 +ω21.

6b) (10 pts) Find z(t).

4

6c) (10 pts) Let y(t) be the solution of the forced equation (4) with initial conditionsy(0) = 0, y′(0) = 0, and let Y (s) be its Laplace transform. Prove that

Y (s) =s

(s2 + ω2) [(s+ p)2 + ω21 ]

(5)

Do not attempt to perform partial fraction separation.

5

6d) (10 pts) Using the partial fraction expansion technique, we have

Y (s) =1

(ω20 − ω2)2 + (2pω)2

[(ω2

0 − ω2) · s + 2pω2

s2 + ω2− (ω2

0 − ω2) · (s+ p) + p(ω2 + ω20)

(s+ p)2 + ω21

].

Do not attempt to prove this. For this question, simply indicate which part of thisexpression corresponds to the steady-state solution and which corresponds to the transientsolution. State why.

6e) (10 pts) Use the table of Laplace transforms to prove that the steady-state solution is

y(t) =

[ω20 − ω2

(ω20 − ω2)2 + (2pω)2

cos(ωt) +2 pω

(ω20 − ω2)2 + (2pω)2

sin(ωt)

]u(t). (6)

6

6f) (20 pts) Using the result from question (6e), show that the total amplitude of thesteady-state solution as a function of ω is

A(ω) =1√

(ω20 − ω2)2 + (2pω)2

. (7)

Assuming ω0 = 2 and p = 12 , sketch the frequency response plot. Indicate clearly any critical

points.

7

Part III (100 points)

Problem 7. (25 points total) Consider the following non-linear system.

dx

dt= −

(1 − x

6

)x +

1

2xy

dy

dt=(

1 − y

4

)y − 1

3xy

(8)

7a) (5 pts) If interpreted as a predator-prey system, which is the prey and which is thepredator?

7b) (5pts) In the absence of any prey species, what type of equation does the predatorspecies follow?

7c) (5pts) In the absence of any predator species, what type of equation does the preyspecies follow?

8

7b) (10 pts) Determine all critical points of the system (8).

9

Problem 8. (40 points total) Consider again the system

dx

dt= −

(1 − x

6

)x +

1

2xy

dy

dt=(

1 − y

4

)y − 1

3xy

(9)

8a) (20 pts) Linearize the system at the critical point (x, y) = (2, 43 ). Determine all eigen-values. Sketch the graph of the linearized system, including any straight-line solutions.

10

8b) (20 pts) Linearize the system at the critical point (x, y) = (6, 0). Determine all eigen-values. Sketch the graph of the linearized system, including any straight-line solutions.

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Problem 9. (35 points total) Consider again the system

dx

dt= −

(1 − x

6

)x +

1

2xy

dy

dt=(

1 − y

4

)y − 1

3xy

(10)

Most predator-prey systems are not Hamiltonian. However, set

H(x, y) = x · y ·(x

6+y

4− 1

). (11)

9a) (5 pts) Verify that the system (10) is a Hamiltonian system.

9b) (5 pts) Prove that H is a Hamiltonian for this system.

12

9c) (10 pts) Sketch the level-sets of H in the first quadrant. To do this, make a precisesketch of the level-set H = 0, then make reasonable approximations for the level-sets H = αfor values of α both larger and smaller than 0.

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9d) (10 pts) Sketch the trajectories of the system (10) in the phase plane. Restrict your dia-gram to the first quadrant. Clearly label all critical points by type, and label any separatricesyou find.

9e) (5 pts) What is the ultimate fate of the system with initial condition (x0, y0) = (1, 1)?What if initial conditions are (x0, y0) = (10, 10)?

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Extra Credit Problem. (20 points total) Consider the following slightly modified versionof (8):

dx

dt= −7

8

(1 − x

7

)x +

15

32xy

dy

dt=(

1 − y

4

)y − 1

3xy

(12)

and again set

H(x, y) = x · y ·(x

6+y

4− 1

). (13)

a) (10 points) Show that the function H is a Lyapunov function for (12), when the systemis restricted to x ≥ 0, y ≥ 0. It may help to rewrite the first equation as

dx

dt=

[−(

1− x

6

)x +

1

2xy

]− x

8

[y4

+x

3− 1

](14)

b) (10 points) On a scratch page, sketch the trajectories in the phase plane for this system.What is the ultimate fate of a system with initial conditions (x0, y0) = (1, 1)?

15

Scratch

16

Scratch

17

Scratch

18

Trigonometric Formulas

Basic formulas

cos(α± β) = cos(α) cos(β) ∓ sin(α) sin(β)

sin(α± β) = sin(α) cos(β) ± cos(α) sin(β)

From these follow the sum-to-product formulas:

cos(α− β) + cos(α+ β) = 2 cos(α) cos(β)

cos(α− β) − cos(α+ β) = 2 sin(α) sin(β)

sin(α+ β) + sin(α− β) = 2 sin(α) cos(β)

sin(α+ β) − sin(α− β) = 2 cos(α) sin(β)

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Laplace Transforms

Definition

F (s) =

∫ ∞0

e−stf(t) dt

Table of Common Transforms

t-domain function s-domain function

δ(t), δa(t) 1, e−as

u(t), ua(t)1s ,

e−as

s

t u(t) 1s2

tn u(t) n!sn+1

eat u(t) 1s− a

sin(ωt)u(t) ωs2 +ω2

cos(ωt)u(t) ss2 +ω2

eat sin(ωt)u(t) ω(s−a)2 +ω2

eat cos(ωt)u(t) (s−a)(s−a)2 +ω2

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Properties of the Laplace Transform

Propertyt – domain function s – domain function

Name

f(t) F (s) =∫∞

0 e−stf(t) dt

Linearity α f(t) + β g(t) αF (s) + β G(s)

dfdt

sF (s) − f(0)∫f 1

s F (s)

t f(t) −dFds

1t f(t) −

∫F

Time Shiftf(t − a) eas F (s)

Theorem

Phase Shifte−at f(t) F (s+ a)

Theorem

Time Scalef(at) 1

|a| F ( sa)Theorem

Convolution(f ∗ g)(t) F (s)G(s)

Theorem

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Convolutions

Definition

(f ∗ g)(t) =

∫ ∞−∞

f(t − u) g(u) du

Property Expression

Symmetry f ∗ g = g ∗ f

Associativity f ∗ (g ∗ h) = (f ∗ g) ∗ h

Linearity f ∗ (αg + βh) = α(f ∗ g) + β(f ∗ h)

Differentiation ddt(f ∗ g) = df

dt ∗ g = f ∗ dgdt

Antidifferentiation∫

(f ∗ g) =∫f ∗ g = f ∗

∫g

The Identity f ∗ δ = f

Integration f ∗ u =∫ t−∞ f(v) dv

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