Practice Exam 2

NAME: Exam 2 Answer Sheet for Problems 1 through 6 1. y(x) = 2. y(x) = 3. y(x) = 4. yp (x) = 5. yp (x) = 6. yp (x) = 1 2 1. The roots and their multiplicities to the characteristic equation of some homogeneous linear ODE with constant coefficients are given below. Find the general solution to the ODE. r1 = −1, k1 = 2 r2 = −2, k2 = 3 2. The roots and their multiplicities to the characteristic equation of some homogeneous linear ODE with constant coefficients are given below. Find the general solution to the ODE. r1 r2 r3 r4 = = = = 2 + 2i, k1 = 1 2 − 2i, k2 = 1 3i, k3 = 2 −3i, k4 = 2 3. The roots and their multiplicities to the characteristic equation of some homogeneous linear ODE with constant coefficients are given below. Find the general solution to the ODE. r1 r2 r3 r4 = = − = 3i, k1 = 1 −3i, k2 = 1 3, k3 = 1 −3, k4 = 1 3 4. The roots and their multiplicities to the characteristic equation of the homogeneous version of some nonhomogeneous linear ODE with constant coefficients are given below, and the function f (x) that makes it nonhomogeneous is provided as well. Find the form of the particular solution. r1 = 1, k1 = 2 f (x) = x3 5. The roots and their multiplicities to the characteristic equation of the homogeneous version of some nonhomogeneous linear ODE with constant coefficients are given below, and the function f (x) that makes it nonhomogeneous is provided as well. Find the form of the particular solution. r1 = −1, k1 = 3 r2 = 0, k2 = 2 f (x) = e−x 6. The roots and their multiplicities to the characteristic equation of the homogeneous version of some nonhomogeneous linear ODE with constant coefficients are given below, and the function f (x) that makes it nonhomogeneous is provided as well. Find the form of the particular solution. r1 r2 r3 r4 f (x) = = = = = 2 + 3i, k1 = 2 2 − 3i, k2 = 2 3 + 2i, k3 = 1 3 − 2i, k4 = 1 10xe−2x cos(3x) 4 7. Find the GENERAL SOLUTION to the second-order nonhomogeneous linear ODE with constant coefficients using Method of Undetermined Coefficients. y ′′ + 3y ′ + 2y = e−2x (a) Find the characteristic equation. Then, find the roots to the characteristic equation and their multiplicities. Use this to find the general solution to the homogeneous version of the ODE, yH (x). (2 pt) (b) Find m, α, and k from f (x) and the roots and multiplicities from earlier. Use these to find the form of the particular solution, yp (x). (2pt) (c) Find the derivatives of yp (x). (2pt) 5 (d) Determine the “undetermined” coefficients in your form of your particular solution by plugging yp (x) and its derivatives into the ODE and solving. (3pt) (e) Write the general solution to the nonhomogeneous equation below, y(x). (1pt) 6 8. Find the GENERAL SOLUTION to the second-order nonhomogeneous linear ODE with constant coefficients using Variation of Parameters. y ′′ + 3y ′ + 2y = e−2x (a) Use the y1 (x) and y2 (x) you found in problem 7. Find u1 (x). (2pt) (b) Find u2 (x). (2pt) 7 (c) Find yp (x). Remember to simplify and combine like terms. (2pt) (d) Find y(x). Remember to simplify and combine like terms. (1pt) 8 BONUS! Consider the nonhomogeneous linear second order ODE with NONCONSTANT coefficients below. x2 y ′′ + xy ′ − y = x2 This is known as a Cauchy-Euler equation and has a similar method to find the homogeneous solutions, which are y1 (x) = x−1 and y2 (x) = x. Use Variation of Parameters 2 to show that y(x) = x3 is a particular solution to the Cauchy-Euler equation. Note that a2 (x) = x2 now! 9