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MAY 20, 2004
Name:
Problem 1: / 10
Problem 2: / 20
Problem 3: / 25
Problem 4: / 15
Problem 5: / 20
Problem 6: / 25
Problem 7: / 10
Problem 8: / 35
Problem 9: / 40
Problem 10: / 10 Extra credit
Total: / 200
Instructions: Please write your name at the top of every page of the exam. The exam is closed book, closed notes, and calculators are not allowed. You will have approximately 3 hours for this exam. The point value of each problem is written next to the problem – use your time wisely. Please show all work, unless instructed otherwise. Partial credit will be given only for work shown.
You may use either pencil or ink. If you have a question, need extra paper, need to use the restroom, etc., raise your hand.
Date : Spring 2004. 1
Table of Laplace Transforms
y(t) Y (s) = L[y(t)]
. y(n)(t) snY (s) − (y(n−1)(0) + · · · + sn−^1 y(0)) . tn^ n!/sn+ . tny(t) (−1)nY (n)(s) . cos(ωt) s/(s^2 + ω^2 ) . sin(ωt) ω/(s^2 + ω^2 ) . eaty(t) Y (s − a) . y(at), a > (^0) a^1 Y (s/a) . S(t − t 0 )y(t − t 0 ), t 0 ≥ 0 e−st^0 Y (s) . δ(t − t 0 ), t 0 ≥ 0 e−st^0 . (S(t)y) ∗ (S(t)z) Y (s)Z(s) . y(t), y(t + T ) = y(t) (^1) −e^1 −sT
� T
0 e
−sty(t)dt
Name: Problem 1, contd.
(b)(5 points) Introduce new variables v 1 = x� 1 and v 2 = x 2 �^. Find a system of 1st^ order linear ODEs satisfied by x 1 , v 1 , x 2 and v 2 of the form, ⎡ ⎤ ⎡ ⎤ x� 1 x 1 ⎢ ⎢ ⎣
v� 1 x� 2
⎦ =^ B^
v 1 x 2
v� 2 v 2
In other words, find the matrix B.
Extra credit(2 points) What is the relationship of pA(λ) and pB (λ)?
Name: Problem 2: / 20
Problem 2(20 points) Consider the ODE,
y(t)�^ +
y(t) = 3e−t
(^3) / 3 , t > 0 t
(a)(5 points) Find an integrating factor.
(b)(10 points) Find the general solution.
(c)(5 points) Find the unique solution that has an extension to a continuous function on [0, ∞).
Name: Problem 3, contd.
(b)(15 points) Use variation of parameters to find a particular solution of the inhomogeneous ODE,
2 t 1 y��^ + t^2 − 4
y�^ − 16 (t^2 − 4)^2
y = 1.
Name: Problem 4: / 15
Problem 4(15 points) Using the method of undetermined coefficients and the exponential shift rule, find a particular solution of the inhomogeneous linear 2nd^ order ODE,
y��^ + 5y�^ + 6y = − 4 te−^3 t^.
Name: Problem 5, contd.
(b)(10 points) An orthonormal basis for the even periodic functions of period 4 is,
1 1 φ 0 (t) = 2
, φn(t) = √ 2
cos(nπt/2), n = 1, 2 , 3 ,...
Compute the Fourier coefficients, � (^2) an = � f , φ� n� = f (t)φn(t)dt, − 2 and express your answer as a Fourier cosine series,
f (t) =
a 0
∞ (^) a n (^) cos(nπt/2). 2
n=1^2
Don’t forget to compute a 0.
Extra credit(3 points) Plug in t = 0 to get a formula for the series,
� (^1) (2m + 1)^2
∞
m=
Name: Problem 6: / 25
Problem 6(25 points) Let f (t) be the piecewise continuous function,
0 , 0 < t < 1 f (t) = e−3(t−1), t ≥ 1
Let y(t) be the continuously differentiable and piecewise twicedifferentiable solution of the following IVP, (^) ⎧ ⎨ y��^ + 5y�^ + 6y = f (t), y(0) = 0, ⎩ (^) y�(0) = 0
Denote by Y (s) the Laplace transform,
L[y(t)] =
∞ e−st^ y(t)dt. 0
(a)(5 points) Compute the Laplace transform of the IVP and use this to find an equation that Y (s) satisfies.
(b)(10 points) Solve the equation fo Y (s) and find the partial fraction decomposition of your answer. Use the Heaviside coverup method to simplify the partial fraction decomposition.
Name: Problem 7: / 10
Problem 7 Let A be the real 3 × 3 matrix, ⎡ ⎤ 2 1 0 A = ⎣^0 − 1 1 ⎦. 0 0 2
(a)(3 points) Compute the characteristic polynomial pA(λ) = det(λI − A).
(b)(7 points) For each eigenvalue, find an eigenvector (not a generalizaed eigenvector ).
Name: Problem 7, contd.
Extra credit(3 points) For one of the eigenvalues, the eigenspace is deficient. Find a generalized eigenvector that is not an eigenvector.
Name: Problem 8, contd.
(c)(5 points) Find the general solution of the linear system of ODEs. Write your answer in the form of a solution matrix X(t) whose column vectors are a basis for the solution space.
(d)(5 points) Compute the exponential matrix,
exp(tA) = X(t)X(0)−^1.
Name: Problem 8, contd.
(e)(10 points) Denote by f (t) the vectorvalued function,
f (t) = (^0)
t .
Denote by x 0 the column vector, 0 x 0 = 1.
For the following IVP write down the solution in terms of the matrix exponential.
x�^ = Ax + f (t), x(0) = x 0.
Compute the entries of the constant term vector and the integrand column vector, but do not evaluate the integrals.
Name: Problem 9, contd.
(c)(15 points) For each linearization, determine the eigenvalues. If the eigenvalues are complex conjugates, determine the rotation (clockwise in/out, counterclockwise in/out). If the eigenvalues are real, determine roughly the eigenvectors and the type of the local phase portrait.
Name: Problem 9, contd.
For the following 2 parts, please sketch your answer on the graph on the following page.
(d)(5 points) Using a dashed line, sketch the xnullcline and ynullcline. Draw a few representative arrows indicating the direction of the orbits on the nullcline on each side of each equilibrium point.
(e)(10 points) Sketch the phase portrait. Label all equilibrium points. For each equilibrium point, sketch a few orbits. In particular, for each saddle sketch each orbit whose limit or inverse limit is the equilibrium point.
There is one basin of attraction. Use bold lines to indicate each (rough) separatrix bounding this basin of attraction. Your sketch should just be a rough sketch, but it should be qualitatively correct.
