MATH 225 Final Exam Practice Problems

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Math 225 Final Exam Practice Problems N. Emerson The Final Exam will be cumulative. However there will be some emphasis on material since Exam 2: Chapters 6 and 7; equivalently Homework 9–11. Revised 12/6/12. 1. Solve x dx + ye− x dy = 0: 2. Solve dy dx + y sec 2 2 x = sec x. 3. Find the general solution of (D 1) 3 2 2 (D + )(D + 225)Dy = 0: 4. Solve the initial-value problem 4y′′ + 4y′ + 4y = 0; y(0) = 3; y′(0) = 7. 5. Solve y′′ + 2y′ + 3y = 1 sin x: 6. Solve the initial-value problem dx dt = [ 2 4 1 3 ] x; x(0) = [ 7 5 ] : 7. Solve the system of equations x′ = [ 3 18 2 9 ] x: 8. Let A be a 4 4 matrix with eigenvalues 1 = 2 + 3i, 2 = 2 3i, 3 = 1 and 4 = 1 with corresponding eigenvectors v1 =   1 + i 4 i 0 0   , v2 =   1 i 4 + i 0 0   , v3 =   0 0 1 0   , and v4 =   0 0 0 1   . Find the general solution of the system of diﬀerential equations x′ = Ax. 9. Find the general solution of the system of equations x′ = [ 6 3 2 1 ] x + [ et 2t e ] 10. Consider the diﬀerential equation y′′ + p(t)y′ + q(t)y = 0; This study resource was shared via CourseHero.com where p and q are continuous for all real numbers. (a) Show that there is a solution f such that f(0) = 1 and f′(0) = 0. (b) Show that there is a solution g such that g(0) = 0 and g′(0) = 1. (c) Show that f and g are linearly independent.

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