Math 123, Spring 2016, Final Exam (version 2) Differential Equations and Linear Algebra Date: Time: Lecture Section: Instructor:
Wednesday, May 18 12:15-2:30 p.m. 001 Matthew Johnston
Last Name: First Name: SJSU Student ID Number:
FOR EXAMINERS’ USE ONLY
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Instructions
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1. Fill out this cover page completely. 2. Answer questions in the space provided, using scratch paper for rough work. 3. Show all the work required to obtain your answers. 4. No calculators are permitted but you may consult a one page hand-written cheat sheet.
Math 123 Final Exam, Spring 2016
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1. Definitions and Classification: [1]
(a) State what it means for a set of vectors {v1 , v2 , . . . , vn } to be linearly independent.
[1]
(b) State a condition on an n × n matrix A which guarantees it is invertible (i.e. A−1 exists).
[1]
(c) Convert the following second-order system into a system of first-order differential equations in the variables x1 (t) = x(t), x2 (t) = x0 (t): x00 + x2 = 2 ln(x + t).
[4]
2. True/False: (a) Consider an n × n matrix A such that det(A) = 2. Then det(A−1 ) = −2. [True / False] (b) Any linearly independent set of vectors {v1 , . . . , vn } where vi ∈ Rn for i = 1, . . . , n, forms a basis of Rn . [True / False]
(c) The integrating factor for the first-order linear system µ(x) = cos(x). [True / False]
dy − tan(x)y = sin(x) is dx
(d) The amplitude of the solution x(t) of the forced pendulum model (mx00 + kx = cos(ωt)) depends on the forcing frequency, ω. [True / False]
Math 123 Final Exam, Spring 2016
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3. Linear Algebra: Consider the following matrix
2 −2 −2 5 . A = −3 1 1 1 −3 [1]
(a) Show that the vector v = (2, 1, 1) is in the null space of A.
[1]
(b) Show that the vector v = (−1, 1, 0) is an eigenvector of A. Determine the coresponding eigenvalue λ.
[2]
(c) Find the determinant of A, det(A).
[2]
(d) If x is a solution to A · x = b and v ∈ null(A), then A · (x + v) = b. Use this observation to find two distinct solutions to the following system: 2x − 2y − 2z = −4 −3x + y + 5z = 4 x + y − 3z = 0 [Hint: You can also solve this system directly, but it is quicker to use the observation in conjunction with the results of (a) and (b)!]
Math 123 Final Exam, Spring 2016
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4. Linear Algebra: Consider the set of vectors S = {v1 , v2 , v3 } where v1 = (−1, 3, 1, −2), v2 = (0, 2, 1, 1), and v3 = (−1, 1, 0, −3).
[3]
(a) Determine if the set of vectors in S are linearly independent or linearly dependent. [Hint: Use the definition of linear independence.]
[3]
(b) Determine the value of k ∈ R such that w = (1, 1, 1, k) is in the subspace W = span(S).
Math 123 Final Exam, Spring 2016
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5. Slope Fields and Solutions: Consider the following initial value problem: dy = (1 − y)(1 + y) dx y(0) = 0
(1)
[3]
(a) Sketch the slope field of (2) over the range −2 ≤ x ≤ 2, −2 ≤ y ≤ 2. [Hint: Since the right-hand-side only depends upon y, a table will suffice!]
[2]
(b) Show that y(x) =
[1]
(c) Describe the behavior of the solution as x → ∞. In particular, does the solution converge to a specific value, or approach some sort of identifiable behavior?
e2x − 1 is a solution of the initial value problem (2). e2x + 1 [Hint: Remember to check the initial condition!]
Math 123 Final Exam, Spring 2016
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6. First-Order Differential Equations:
[3]
[3]
Solve the following differential equations (including initial conditions if specified): y2 − 1 dy = (a) dx 2xy y(1) = 2
(b)
dy y y 2 = − dx x x
Math 123 Final Exam, Spring 2016
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7. Second-Order Differential Equations: Consider the following second order differential equation (derivatives with respect to x): 9y 00 − y = g(x)
(2)
[2]
(a) Determine the complementary solution yc (x) of (2).
[2]
(b) Determine the trial function form yp (x) for (2) given the following g(x) (you do not need to solve for the constants!): 1 (i) g(x) = x sin(2x) 3
1
(ii) g(x) = xe 3 x
[2]
(c) Determine the particular solution yp (x) for (2) for the following g(x) (now you have to solve for the constants!): g(x) = 10 cos(x) + 8ex
Math 123 Final Exam, Spring 2016
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8. Laplace Transforms:
[1]
[1]
[4]
(a) Compute the following Laplace transforms and inverse Laplace transforms: (i) L e−x sin(3x)
(ii) L
−1
s+4 s2 + 4
(b) Use the Laplace Transform method to solve the following inital value problem: 00 y − 4y = 4e2x y(0) = 0 0 y (0) = 1
Math 123 Final Exam, Spring 2016
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9. Linear Systems of Differential Equations: Consider the following first-order system of differential equations: dx = −x + 2y x(0) = 3 dt dy = 4x + y y(0) = 3 dt [3]
(a) Sketch the slope field of (3) in the (x, y)-plane.
[4]
(b) Solve the initial value problem (3) for x(t) and y(t).
(3)
Math 123 Final Exam, Spring 2016
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THIS PAGE IS FOR ROUGH WORK
