Math 256, Introduction to Differential Equations: Homework

• 9/27    1.2. 1, 2, 7, 11

• 9/29    2.1. 1c), 3c), 13-20, 31, 34
             2.2.  3, 7, 8, 9, 14, 19, 23 

• 10/1    2.3.  3, 7, 8, 13, 23

  Homework due Wednesday, Oct. 6

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• 10/4    2.3. 9, 10

• 10/6    2.4. 1-6, 22a),b), 25, 27, 28

  Homework due Wednesday, Oct. 13

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• 10/11    2.5. 1, 3, 5, 6, 7, 9, 10, 15a), 16, 17a)
• 10/13    2.5. 20, 23
• 10/15    2.7.  1, 4, 11a), 12a)

           Homework due Wednesday, Oct. 20

• 10/18    2.7.  5, 7, 15, 16
               1.1.  15-20   

• 10/20    3.1.   1, 3, 5, 7, 9, 10, 16, 17, 20, 23

• 10/22     - Prove Th.2  from class (Superposition principle): If y1, y2 are solutions of the ODE
               y'' + p(t)y' + q(t) = 0, then c1 y1+c2 y2 is a solution for any choice of coefficients c1, c2
               3.2. 1, 3, 5, 6, 7, 8, 10, 12, 13.

  Homework due Wednesday, Oct. 27

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• 10/25      - Prove Ex.2  from class: If the characteristic equation  of the ODE ay''+by'+cy = 0
                 has two different real solution r1, r2, then exp(r1 t), exp(r2 t) form a fundamental set of solutions.
                 3.2. 14, 15, 17,29
                 3.3. 1,  3, 4, 5, 7, 9-12, 20, 23-25 
• 10/27      3.4. 1-6, 7-10, 12-14, 17, 18, 20, 22, 27, 28, 31, 38, 39
• 10/29      - Prove Exercise from class: Assume that the characteristic equation of ay''+by'+cy = 0 has repeated root $r$.
                 Compute W(exp(rt), t exp(rt)).  Show that (exp(rt), t exp(rt)) form a fundamental set of solutions for this differential
                 equation.        

          Homework due Wednesday, Nov. 3

• 11/1        Review session, no homework.

• 11/3        3.6.  1, 2, 3, 6,13,17,18, 31, 32

• 11/5        3.6.  4, 5, 8, 14, 16

          Homework due Wednesday, Nov. 10

• 11/8       4.1   3, 7-10, 20
                4.2   1-61 
                Exercise on determinants: download .pdf  file  from here  
                Prove Principle of Superposition for the 3rd order linear homogeneous ODE 

• 11/10     4.1    12 (you do not have to compute Wronskian for 12), 15

• 11/12     Do two exercises from class.
                Exercise 1: show that tcos(t) and tsin(t) ate solutions of the ODE y''-2y'+y =0
                Exercise 2: compute W(e^t, sin(t), cos(t)).   
                4.2    7-10, 11, 12, 13, 15, 18, 20, 29, 30

          Homework due Wednesday, Nov. 17

• 11/15    No homework
• 11/17    4.3    1-8, 15, 18
• 11/19    Exercise 1 from class. Check that 3(ln|sin t|) sin t + 3t cos t i a solution of y'' + y = 3/sin t.
               Using variation of parameters method (i.e. reduction of order) as in Example 1 from class, find general solution of
                      b) y''' -3y'' + 3y - y = 0
               3.5    23, 24, 28
               3.7    1, 3, 28, 29

          Homework due Wednesday, Dec. 1

• 11/22    3.7    5, 10, 13
               4.4    1, 2, 3
• 11/23    5.1    1, 2, 4, 5, 8, 12, 13, 14 ,15

         Homework due Wednesday, Dec. 1