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CU-Boulder APPM 2360 - Diffeq exam 3g

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APPM 2360 Exam 3 April 14, 2004 1INSTRUCTIONS:• Computers, calculators, books, notes, and crib sheets are not permitted.• Write your name, instructor’s name, and recitation number on the front of your bluebook.• Work all five problems. Start each problem on a new page.• Show your work and clearly identify your final answer.1. (20 points) For the following equationy00− 4y0+ 4y =e2tta. Find the characteristic equation. (5 points)b. Find two linearly independent solutions to the homogeneous equation. (4 points)c. Find the particular solution. (6 points)d. Give the general solution. (2 points)e. Solve the initial value problem for t0= −1, y(t0) = e−2, and y0(t0) = e−2. (3 points)2. (20 points)a. Solve the homogeneous differential equation y00− 2y0+ y = 0. (6 points)b. Find the solution to the above problem satisfying the initial value: y(0) = 2, y0(0) = 5. (4points)c. Predict the most suitable form of ypfor the nonhomogeneous differential equation y00−2y0+ y = tet(DO NOT SOLVE). (4 points)d. Find the particular solution to the equation in part c. (6 points)APPM 2360 Exam 3 April 14, 2004 23. For the system of linear differential equationsx0= AxwhereA =3 1−1 1a. Find all the eigenvalues and corresponding eigenvectors of A. (8 points)b. Find the general solution to x0= Ax. (6 points)c. Find the solution to the above problem satisfying the initial value: x1(0) = 4, x2(0) = −2.(6 points)4. (20 points) Given the matrixA =3 1 00 3 10 0 1.answer the following questions.a. Find the eigenvalues of A (5 points).b. Find the eigenvectors of A (10 points).c. Give the eigenspace associated with each eigenvalue (5 points).5. (20 points) Answer the following true/false questions:a. If y1and y2are two solutions of y00− y0sin(t) + y = 0 then y = c1y1+ c2y2is the generalsolution.b. The vectors u = (1, 1, 1, 2), v = (0, 1, 2, 3), and w = (1, 0, 0, 0) form a basis of R4.c. The equation Ax = b has a solution if and only if b is a scalar multiple of a column in A.d. The solution space of y00+ y0+ y = sin(t) is a 2-dimensional vector space.e. If Ax = x, then x is an eigenvector of


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CU-Boulder APPM 2360 - Diffeq exam 3g

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