MATH 215/255 Homework 7 Submit online via Canvas by 6pm, Thursday, March 17, 2022. This homework set has written problems only. §6.3, §6.4, §3.1, §3.3 The written part counts for 20 points. 1. 6.3.4: Find the solution to mx00 + cx0 + kx = f (t), x(0) = 0, x0 (0) = 0 for an arbitrary function f (t), where m > 0, c > 0, k > 0, and c2 − 4km < 0 (system is underdamped). Write the solution as a definite integral. Hint. Write ms2 + cs + k = m[(s + p)2 + ω12 ] where p = c 2m , ω12 = k m − p2 = 4km−c2 4m2 > 0. 2. Solve x00 + 4x0 + 5x = δ(t − π), x(0) = 0, x0 (0) = 2. 3. Solve x00 + 4x0 + 3x = 2δ(t − π), x(0) = 2, x0 (0) = 0. 4. 3.1.2: Find the general solution of x01 = x2 − x1 + t, x02 = x2 . 5. Rewrite the damped mass-spring system for two masses and three springs m1 x001 + cx01 = −k1 x1 + k2 (x2 − x1 ), m2 x002 + cx02 = −k2 (x2 − x1 ) − k3 x2 as a first order system of ODEs. The constants c, mi and ki are positive. 6. 3.3.1: Write the system x01 = 2x1 − 3tx2 + sin t, x02 = et x1 + 3x2 + cos t in the vector form ~x0 = P (t)~x + f~(t), where P (t) is a 2 × 2 matrix. 7. 3.3.2: Consider linear system 0 ~x = 1 3 3 1 ~x. a) Verify that the system has the following two solutions 1 1 −2t 4t ~x(t) = e , ~x(t) = e . −1 1 b) Show that they are linearly independent and write down the general solution. c) Write down the general solution in the form x1 (t) =?, x2 (t) = ? (i.e. write down a formula for each element of the solution). 8. 3.3.104: a) Write x01 = 2tx2 , x02 = 2tx2 in matrix notation. b) Solve its general solution, and write it in matrix notation. 1