Practice Final, 427K, 05/08/2014  PRINTED NAME:  EID:

No books, notes, calculators, or telephones are allowed.

Every problem is worth an equal number of points. You must show your work; answers without substantiation do not count.

Answers must appear in the box provided! No or the wrong answer in the answer box results in no credit!

This does not aim nor claim to be exhaustive! Use this as a guide of what to study and not of what not to study! Do not expect to find every test problem listed here! Sigh. There will be 7–10 problems on the final. A problem worked out in class or one of the problems assigned from Sections 1.1-2, 2.1-2. Answer:

A problem worked out in class or one of the problems assigned from Sections 2.3-5. Answer:

A problem worked out in class or one of the problems assigned from Sections 2.6-8, 3.1-2. Answer:

A problem worked out in class or one of the problems assigned from Sections 3.1-3.2. Answer:

A problem worked out in class or one of the problems assigned from Sections 3.3-3.6. Answer:

What does the Wronskian do for you?

a) How is it defined: W [y_{1}, y_{2}] =^{def}

b) What does it say about a pair y_{1}, y_{2} of solutions of a SOLODE?

c) State Abel’s theorem.

State the existence and uniqueness theorem for the general FOODE.

State the existence and uniqueness theorem for the SOLODE.

Describe Euler’s one-step approximation method - how do you get from one point to the next?

Describe the method of Integrating Factors. What does it mean to say that ϕ(t) is a solution of y′ = f(t, y)?

Find the general solution of the FOLODE ty′ + 2y = sin t, t > 0.

A corpse is discovered hanging on a hook in a meat locker that is being kept at −3^{∘}C. Its temperature is measured immediately and is found to be 27^{∘}C. Three hours later its temperature is down to 7^{∘}C. Assuming that at the time of death the body had the normal body temperature of 37^{∘}C, when did death occur?

Answer:

The pig was slaughtered hours before discovery.

A certain population Y (t) satisfies the logistic equation Y′ = Y (1 − Y/1000). At time t = 0 its value Y (0) is 30% of the carrying capacity.

(a) Find Y (t)

(b) At which time T does Y (T) reach 60% of the carrying capacity?

Answer: (a): Y (t) = (b): T = Find an integrating factor μ and solve y dx + (2x − ye^{y}) dy = 0.

Answer: μ( ) =  , sol’n:  = c

The FOOIVP

y′ = \frac{1}{1 + t^{2}y^{3}} ,  y(1) = 1 ,

cannot be handled with any of our four methods, so you decide to approximate y(1.2) using Euler’s method. What approximate value for y(1.2) do you get?

Answer: y(1.2) ≈ !!!Read the whole problem before answering!!!

a) Describe the most general circumstance in which the Method of Variation of Parameters applies.

b) What is it intended to accomplish?

c) How does it work?

!!!Read the whole problem before answering!!!

a) Describe the most general circumstance in which the Method of Reduction of Order applies.

b) What is it intended to accomplish?

c) How does it work?

Find a particular solution of y′′ + 4y = 3 csc t. [Hint: ∫ csc t dt = ln | csc t + cot t| + C.] Use Undetermined Coefficients (!) to find the general solution of y′′ + 2y′ + 5y = 3 cos(2t).

Answer: y = Do one of: p190 # 1-21; p203 # 1-12. Answer:

Do one of: p249 # 1-20, 21-28; p259 # 1-14, 16-21; p265 # 1-17. Answer:

Do one of: p271 # 1-14; p278 # 1-22. Answer:

Do one of: p284 # 1-16; p292 # 1-17; p312 # 1-24; p322 # 1-16. Answer:

Find the regular singular points of (1 − 2x^{2})y′′ + xy′ + y = 0. For each one of them predict the convergence radius of its “mucked–up Euler solution.” Answer:

!!!Read the whole question before answering its parts!!! What does the ratio test for power series do for you?

(a) When does it apply and what does it produce?

(b) What exactly does it say?

(c) Can you use it to determine the convergence radius for the cosine series cos x = 1 − x^{2}/2! + x^{4}/4! ± …?

!!!Read the whole question before answering its parts!!! Suppose the functions P, Q, R in the SOLODE P (x)y′′ + Q(x)y′ + R(x)y = 0 are analytic at the point x_{0}.

(a) What does it mean that x_{0} is an ordinary point? Answer: x_{0} is an ordinary point if .

(b) If x_{0} is an ordinary point, what do you do?

(c) What can you say about the convergence radius of the resulting power series?

Complete the following definitions concerning the SOLODE

P (x)y′′ + Q(x)y′ + R(x)y = 0  (∗)

with analytic coefficients P, Q, R:

(a) x_{0} is an ordinary point for (∗) if ;

(b) x_{0} is a singular point for (∗) if ;

(c) x_{0} is a regular singular point for (∗) if !!!Read the whole question before answering its parts!!! Suppose the functions P, Q, R in the SOLODE P (x)y′′ + Q(x)y′ + R(x)y = 0 are analytic at the point x_{0}.

(a) What does it mean that x_{0} is a regular singular point? Answer: x_{0} is a regular singular point if

(b) If x_{0} is an regular singular point, what do you do?

(c) What can you say about the convergence radius of the resulting series?

Ve) Write down the general Euler equation and describe how you would treat it.

Some Laplace Transforms

f(t)  F (s)  f(t)  F (s)
sin at  a/(s^{2} + a^{2})  t^{n}  n!/s^{n+1}
cos at  s/(s^{2} + a^{2})  u_{c}(t)f(t − c)  e^{−cs}F (s)
Sinh at  a/(s^{2} − a^{2})  δ_{c}(t)  e^{−cs}
Cosh at  s/(s^{2} − a^{2})  f(at)  a^{−1}F (s/a)
f′(t)  sF (s) − f(0)  e^{ct}f(t)  F (s − c)
f′′(t)  s^{2}F (s) − sf(0) − f′(0)  f ∗ g(t)  F (s)G(s)

To which signal functions f does the Laplace transform apply?

(b) Write down the definition: L{f}(s) =^{def} , s Compute the Laplace transform of some simple function. Write down the definition of the convolution of two functions f, g. How do convolution and Laplace transform interact?

Do one of: p312 # 1-24; p322 # 1-16; p337 # 1-16; p344 # 1-12, 17-22; p351 # 1, 3-10; p351 # 3-11, 13-18; p322 # 1-26.

For example, find the Laplace–inverse of F (s) =^{def} \frac{e^{−s}}{s^{2}(s^{2} − 1)}. Answer: L^{−1}{F (s)}(t) = Do one of: p344 # 1-12, 17-22; p351 # 1, 3-10; p351 # 3-11, 13-18; p322 # 1-26. Let the function g : [0, ∞) → ℝ be defined by [drawing it might help]

g(t) =^{def} \{ \begin{matrix}t & for 0 ≤ t ≤ 1, \\ 2 − t & for 1 ≤ t ≤ 2, \\ 0 & for 2 ≤ t < ∞. \end{matrix}

Use the Laplace transform to solve the IVP y′′ − y = g, y(0) = 1, y′(0) = 0.

Answer: y =  + \{  \begin{matrix}for 0 ≤ t ≤ 1; \\ for 1 ≤ t ≤ 2. \\ for 2 ≤ t < ∞. \end{matrix}

Apply the Laplace Transform to a CCSOLODE with Impulse Input. A weight of mass 100g stretches a spring 10cm. If the mass is pulled down an additional 3cm and then is released, and if there is no air resistance, determine the displacement u(t) of the mass at any time t. [The gravitational constant in the metric system is roughly 10m/sec^{2}. Remember that 100cm = 1m] Answer: u(t) = [m], time in [sec].

Find the general power series solution of 2y′′ + (x + 1)y′ + 3y = 0 about the point x_{0} = 2.

Solve the SOLIVP x^{2}y′′ + 3xy′ + 5y = 0, x > 0, y(1) = 1, y′(1) = −1.

Answer:

Answer a theoretical question as in part I, problem V.

Show that x^{2}y′′ + xy′ + (x − 2)y = 0 has a regular singular point at x_{0} = 0. Find the indicial equation and its roots, the recurrence relation, and the series solutions for x > 0 corresponding to the two roots (three non–zero terms will suffice).

Answer: y_{1} = y_{2} = Find the Laplace-inverse of F (s) =^{def} \frac{2}{(s − 3)(s^{2} − 4s + 5)}.

Solve y′′ + y = u_{π}, y(0) = 1, y′(0) = 0. Look at the old quizzes. I might put one similar to them on the test.

Describe the Euler method, the improved Euler method, and the Runge–Kutta method, including estimates of the local and global errors in terms of the step size, and the number of computations required.

(a) State Fourier’s theorem. Describe the Gibbs phenomenon. Describe the method of separation of variables. What are even (odd) functions? How can you get a pure sine (cosine) series for a function f : [0, L] → ℝ from Fourier’s theorem? Do one of: p449 #1ab-12ab; p456 # 1-12; p461 # 1-12. Do one of: p610 # 1-6; p575 # 1–21; p585 # 1-24; p592 # 1ab-6ab, 7a-12a; p 600 # 1-26. Let f : [−π, π] → ℝ be the function defined by [drawing it might help]

f(x) = \{ \begin{matrix}−π − x & for −π ≤ x ≤ −π/2 \\ x & for −π/2 ≤ x ≤ π/2 \\ π − x & for π/2 ≤ x ≤ π \end{matrix}

(a) Find the Fourier series f̃ of f. (b) At which points x is f(x) = f̃(x)? (Give reasons)

Answer: (a) f̃(x) = Do one of p620 # 1-8, 9-13; p632 # 1-8; p645 # 1-5; p632 # 1-8 Answer:

Solve the heat conduction problem u_{t} = 7u_{xx} in an insulated rod of length π whose ends are maintained at 0^{∘} Celsius at all times and whose initial temperature u(x, 0) is given by u(x, 0) = f(x) ∀ x ∈ [0, 2π], where

f(x) =^{def} \{ \begin{matrix}x & for 0 ≤ x ≤ π/2, \\ π − x & for π/2 ≤ x ≤ π. \end{matrix}

Do one of: p610 # 1-14; p620 # 1-14; p632 # 1-8.

Do one of: p 7 # 1, 3, 5, 11, 13, 15, 17, 19, 21; p 15 # 1–7; p 24 all; p 39 # 1–30; p 47 # 1–20, 30–38. Answer:

Do one of: p 59 # 1–14. p 99 # 1–22; 25–31; p 88 # 1–4, 16; p107 # 1–4. Answer:

Do one of: p 75 # 1–12; p142 # 1–25; p151 # 1–14. Answer:

Do one of: p158 # 15–21; p164 # 1–25; p172 # 1–15, 20, 22–30. Answer:

Do one of: p184 # 1–26. Answer:

Find the regular singular points of (1 − 2x^{2})y′′ + xy′ + y = 0. For each one of them predict the convergence radius of its “mucked–up Euler solution.” Answer:

!!!Read the whole question before answering its parts!!! What does the ratio test do for you?

(a) When does it apply and what does it produce?

(b) How does it work?

(c) Can you use it to determine the convergence radius for the cosine series cos x = 1 − x^{2}/2! + x^{4}/4! ± …?

Write down the definition of the Laplace transform of a function f(t). To which differential equations does the Laplace transform apply? How is the Laplace transform used to solve differential equations? Write down the definition of the convolution of two functions f, g. How do convolution and Laplace transform interact? Do one of: p311 # 1-24; p320 # 1-16; p329 # 1-17; p337 # 1-16; Do one of: p344 # 1-12, 17-22; p351 # 1, 3-10; p351 # 3-11, 13-18; p320 # 1-26.

Apply the Laplace Transform to a CCSOLODE with Impulse Input.

(a) State Fourier’s theorem.

(b) Describe the Gibbs phenomenon.

(c) Describe the method of separation of variables.

(d) What are even (odd) functions?

(e) How can you get a pure sine (cosine) series for a function f : [0, L] → ℝ from Fourier’s theorem? Do one of: p451 1ab-12ab; p458 # 1-12; p463 # 1-12. Do one of: p618 # 1-6; p593 # 1-24; p608 # 1ab-6ab, 7a-12a; p600 # 1-26; p610 # 7-12 Let f : [−π, π] → ℝ be the function defined by [drawing it might help]

f(t) = \{ \begin{matrix}−π − x & for −π ≤ x ≤ −π/2 \\ x & for −π/2 ≤ x ≤ π/2 \\ π − x & for π/2 ≤ x ≤ π \end{matrix}

(a) Find the Fourier series f̃ of f. (b) At which points x is f(x) = f̃(x)? (Give reasons)

Answer: (a) f̃(x) = I) (a) Apply the Laplace Transform to a CCSOLODE with Impulse Input. I’) (a) State Fourier’s theorem.

(b) Describe the Gibbs phenomenon.

(c) Describe the method of separation of variables.

(d) What are even (odd) functions?

(e) How can you get a pure sine (cosine) series for a function f : [0, L] → ℝ from Fourier’s theorem?

Do one of: p344 # 1-12, 17-22; p351 # 1, 3-10; p351 # 3-11, 13-18. Answer:

Do one of: p449 #1ab-12ab; p456 # 1-12; p461 # 1-12. Answer:

Let f : [−π, π] → ℝ be the function defined by [drawing it might help]

f(t) = \{ \begin{matrix}−π − x & for −π ≤ x ≤ −π/2 \\ x & for −π/2 ≤ x ≤ π/2 \\ π − x & for π/2 ≤ x ≤ π \end{matrix}

(a) Find the Fourier series f̃ of f. (b) At which points x is f(x) = f̃(x)? (Give reasons)

Answer: (a) f̃(x) = Do one of: p585 # 1-24; p529 # 1-12; p 600 # 1-26.

Answer:

Solve the heat conduction problem u_{t} = 7u_{xx} in an insulated rod of length 2π whose ends are maintained at 0^{∘} Celsius at all times and whose initial temperature u(0, x) is given by u(0, x) = f(x) ∀ x ∈ [0, 2π], where

f(x) =^{def} \{ \begin{matrix}x & for 0 ≤ x ≤ π/2, \\ π − x & for π/2 ≤ x ≤ π. \end{matrix}

Do a similar wave equation problem: let

f(x) =^{def} \{ \begin{matrix}x & for 0 ≤ x ≤ π/2, \\ π − x & for π/2 ≤ x ≤ π. \end{matrix}

Solve the wave equation u_{tt} = 81u_{xx} for a string of length π with initial conditions u(0, x) = f(x) and u_{t}(0, x) = 0.

Do a similar Laplace equation problem: Let

f(x) =^{def} \{ \begin{matrix}x & for 0 ≤ x ≤ π/2, \\ π − x & for π/2 ≤ x ≤ π. \end{matrix}

Then solve the Laplace equation on a square sheet of side π with the boundary conditions u(0, y) = u(x, 0) = u(π, y) = 0 and u(x, π) = f(x).

Do one of: p610 # 1-14; p620 # 1-14; p632 # 1-8.