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FACULTY OF ARTS & SCIENCESUniversity of Toronto

MAT234: Differential Equations

Final Exam, Winter 2017

Duration: 2.5 hoursTotal: 200 points

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Warning! This Test May ONLY Be Written in PEN. Not writing in pen isan automatic deduction of 10 points.

There are 11 questions in total.

DO NOT REMOVE any pages from the test booklet. If you happen to needto use the last page CLEARLY INDICATE on whatever question you’reworking on that it is continued on the extra page.

Do NOT begin until you are instructed to do so.

When you are told the test has ended you MUST stop writing at once.Failure to do so is an academic offence.

Electronic Aids of any kind are Forbidden.

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Final Exam, Winter 2017 MAT234 - Differential Equations

Problem 1. (15 points total). Consider the equation

ye2xy + x+Mxe2xydy

dx= 0, (1)

where M is a number.

a. (5 points). Find the value of M for which the equation (1) becomes exact.

b. (10 points). Solve equation (1) with the value of M you found in part (a) for y(x).

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Problem 2. (10 points). Find the general solution to

ty′(t) + 2y(t) = sin t, t > 0

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Problem 3. (17 points). Let f(x) = x2 for 0 ≤ x ≤ 2, and let g(x) be the odd periodic extension off(x) of period 4.

a. (2 points). Find g(x) and sketch its graph.

b. (5 points). Compute the coefficients of the Fourier sine series for g(x).

c. (5 points). Compute the coefficients of the Fourier cosine series for g(x).

d. (2 points). Write the Fourier series for g(x).

e. (3 points). Sketch the graph of the Fourier series for g(x).

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(extra paper for Problem 3 )

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Problem 4. (35 points total). A mass is attached to a spring, initially at rest at its equilibriumposition. The mass-spring system is frictionless; however, a forcing of 0.25 cos(2t) is applied. Inthe absence of an applied force, the system would oscillate at a natural frequency of 1.5 rad/s.

a. (5 points). Let y(t) be the displacement of the mass from equilibrium. Write an initial valueproblem which describes y(t).

b. (10 points). Solve the initial value problem in part (a) for y(t).

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c. (5 points). Write the solution y(t) you found in part (b) in phase-amplitude form.

d. (5 points). Write the initial value problem in part (a) as a system of the form x′(t) = Ax(t) +b(t), x(0) = x0, where A is a constant coefficient matrix and b(t), x(t), and x0 are vectors.

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e. (5 points) Without using eigenvalues or eigenvectors, find the solution x(t) to the problemin part (d). If you use eigenvalues/eigenvectors you will receieve a score of zero on this partand the next

f. (5 points). Draw a phase diagram for the system in part (d).

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Problem 5. (15 points). Find all eigenvalues and eigenfunctions for the boundary value problem

y′′ + λy = 0,

y′(0) = y(L) = 0.

To receive full marks you must justify your answer.

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Problem 6. (25 points). Consider the heat equation boundary value problem

∂u

∂t=

∂2u

∂x2− 2

u(0, t) = 0

u(1, t) = 1

u(x, 0) = x2 − x+ 2.

a. (8 points). Solve for the steady-state solution v(x).

b. (6 points). Definew(x, t) := u(x, t)−v(x). Write down a boundary value problem forw(x, t).

c. (8 points). Use separation of variables in part (b) to solve forw(x, t) as a series solution. If thisrequires calculating Fourier coefficients or other integrals, you must do those calculationsto receive full marks.

d. (3 points). Give the solution u(x, t) to the original problem.

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(extra paper for Problem 6)

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Problem 7. (12 points total, 3 points each). On the next page, you’re given a collection of linearsystems of ODEs. Below are four phase portraits. Above each phase portrait, write the letter ofthe corresponding linear system.

This is system: This is system:

This is system: This is system:

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Consider the system x′ = Ax where the matrices A are given below.

system letter

A A =

(−2 −4−1/2 −1

)λ1 = −3, v⃗1 =

(41

)λ2 = 0, v⃗2 =

(−21

)B A =

(−2 5−1 2

)λ1 = −i, v⃗1 =

(2 + i1

)λ2 = i, v⃗2 =

(2− i1

)C A =

(−13/8 3/41/4 −1/4

)λ1 = −7/4, v⃗1 =

(6−1

)λ2 = −1/8, v⃗2 =

(12

)D A =

(−1 41/2 −2

)λ1 = 0, v⃗1 =

(41

)λ2 = −3, v⃗2 =

(−21

)E A =

(2 00 2

)λ1 = 2, v⃗1 =

(11

)λ2 = 2, v⃗2 =

(−11

)F A =

(2 −51 −2

)λ1 = i, v⃗1 =

(2 + i1

)λ2 = −i, v⃗2 =

(2− i1

)G A =

(−1 00 −1

)λ1 = −1, v⃗1 =

(10

)λ2 = −1, v⃗2 =

(01

)H A =

(−1/4 −3/4−1/4 −13/8

)λ1 = −1/8, v⃗1 =

(6−1

)λ2 = −7/4, v⃗2 =

(12

)

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Problem 8. (10 points). After being shaken as a young cub1 in a bizarre science experiment2, Tonythe Siberian Tiger grew up to be relatively normal. Tony, having experienced an exciting life inthe lab, never truly fit in in the wild. Ennui set in. To cope, he began practising the bouzouki3

in the hopes of winning the favour of his former scientist caretaker, recalling she also played thebouzouki for him when he was in her lab. Being a remarkably clever tiger, he decided to tunehis bouzouki so that it would “speak with the same voice” as hers: the first string on his bouzoukiwould have the same natural frequencies as the first string on hers and so on. Late one night,Tony snuck into her lab and performed his masterpiece “A Tiger’s Heart Must Roar On” for her.Needless to say, she was quite impressed and readopted him.

The strings on Tony’s bouzouki were 3 times the length of the strings on the scientist’s bouzouki.They’re made out of exactly the same material. How should the tension in the strings of Tony’sbouzouki be related to the tension in the strings of scientist’s bouzouki?

1No actual tigers were harmed in the writing of this exam2The result of which, strangely, earned the maniacal lab partner both a felony conviction and a Nobel Prize, having

unwittingly discovered a new source of energy....3Bouzouki is a stringed Greek instrument, similar to a guitar. If you’re wondering where Tony found such a thing

in the jungle stop thinking so hard and go back to the exam!!

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Problem 9. (20 points total, 5 points each). These are multiple choice problems with no partialcredit.

(i) Consider the heat conduction problem:ut = α2 uxx for all x ∈ (0, L), t > 0

u(0, t) = 0 for all t > 0

u(L, t) = 0 for all t > 0

(2)

One of the following series is the general solution of this problem. Circle it.

∞∑n=1

cn e−(αnπ/L)2t sin(nπx/L)

∞∑n=1

cn e−α(nπ/L)2t sin(nπx/L)

∞∑n=2

cn e−(αnπ/L)2t sin(nπx/L)

∞∑n=1

cn e−(αnπ/L)2t sin(n2πx/L)

(ii) Consider the heat conduction problem:ut = α2 uxx for all x ∈ (0, L), t > 0

ux(0, t) = 0 for all t > 0

ux(L, t) = 0 for all t > 0

(3)


∞∑n=1

cn e−(αnπ/L)2t cos(nπx/L)

∞∑n=1

cn e−α(nπ/L)2t cos(nπx/L)

∞∑n=0

cn e−(αnπ/L)2t cos(nπx/L)

∞∑n=1

cn e−(αnπ/L)2t cos(n2πx/L)

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(iii) Consider the heat conduction problem:ut = α2 uxx for all x ∈ (0, L), t > 0

ux(0, t) = 0 for all t > 0


(4)


∞∑n=1

cn e−(α 2n+1

2πL)2t cos

(2n+ 1

2πx

L

) ∞∑n=1

cn e−(α 2n+1

2πL)2t sin

(2n+ 1

2πx

L

)∞∑n=0

cn e−(α 2n+1

2πL)2t cos

(2n+ 1

2πx

L

) ∞∑n=0

cn e−α( 2n+1

2πL)2t cos

(2n+ 1

2πx

L

)(iv) Consider the wave propagation problem:

utt = a2 uxx for all x ∈ (0, L), t > 0

u(0, t) = 0 for all t > 0


(5)


∞∑n=1

(cn cos(

anπ

Lt) + kn sin(

anπ

Lt))

sin(nπx

L)

∞∑n=1

(cn cos(

an2π

Lt) + kn sin(

an2π

Lt)

)sin(n2π

x

L)

∞∑n=2

(cn cos(

anπ

Lt) + kn sin(

anπ

Lt))

sin(nπx

L)

∞∑n=1

cn cos(anπ

Lt) cos(nπ

x

L) + kn sin(

anπ

Lt) sin(nπ

x

L)

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(extra paper for Problem 9)

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Problem 10. (20 points total). This question has four parts. Consider the wave equation

utt = c2uxx x ∈ R, t > 0

(i) (8 points). Show that u(x, t) = F (x − ct) + G(x + ct) will always solve the above PDE forarbitrary twice-differentiable functions F , G defined on R.

(ii) (4 points). Does this mean that 17 − sin2(x + ct) is a solution to the above wave equation?Why or why not?

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(iii) (4 points). Does this mean that e17 sinh(ex−ct) − 23(x + ct)π is a solution to the above wave

equation? Why or why not?

(iv) (4 points). Does this mean that√

2 + sin(x2 − c2t2) is a solution to the above wave equation?Why or why not?

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Problem 11. (21 points total, 3 points each) For the following questions, answer using the word“True” or the word “False”. You don’t need to justify your answer to receive full credit. However,to discourage guessing we will deduct 2 points for each incorrect answer given.

(i) First order, homogeneous linear systems of ordinary differential equations are always solv-able.

(ii) Boundary value problems for second order, linear, homogeneous ordinary differential equa-tions are always solvable.

(iii) Boundary value problems for second order, linear, homogeneous ordinary differential equa-tions are never uniquely solvable.

(iv) Boundary value problems for second order, linear, homogeneous ordinary differential equa-tions are always uniquely solvable when they happen to be solvable.

(v) Suppose a, b are fixed numbers. If a function f ∈ PC[a, b] is orthogonal to all constants,then f ≡ 0.

(vi) Suppose that f is a continuous function on [0, L]. The the Fourier Sine Series of f will becontinuous on R

(vii) If u(x, t) solves the one-dimensional wave equation with wave speed c = 1, then so doesu(t, x).

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(extra paper )

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