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8.7E: Constant Coefficient Equations with Impulses (Exercises)

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Q8.7.1

In Exercises 8.7.1-8.7.20 solve the initial value problem. Graph the solution for Exercises 8.7.2, 8.7.4, 8.7.9, and 8.7.19.

1. y+3y+2y=6e2t+2δ(t1),y(0)=2,y(0)=6

2. y+y2y=10et+5δ(t1),y(0)=7,y(0)=9

3. y4y=2et+5δ(t1),y(0)=1,y(0)=2

4. y+y=sin3t+2δ(tπ/2),y(0)=1,y(0)=1

5. y+4y=4+δ(t3π),y(0)=0,y(0)=1

6. yy=8+2δ(t2),y(0)=1,y(0)=1

7. y+y=et+3δ(t6),y(0)=1,y(0)=4

8. y+4y=8e2t+δ(tπ/2),y(0)=8,y(0)=0

9. y+3y+2y=1+δ(t1),y(0)=1,y(0)=1

10. y+2y+y=et+2δ(t2),y(0)=1,y(0)=2

11. y+4y=sint+δ(tπ/2),y(0)=0,y(0)=2

12. y+2y+2y=δ(tπ)3δ(t2π),y(0)=1,y(0)=2

13. y+4y+13y=δ(tπ/6)+2δ(tπ/3),y(0)=1,y(0)=2

14. 2y3y2y=1+δ(t2),y(0)=1,y(0)=2

15. 4y4y+5y=4sint4cost+δ(tπ/2)δ(tπ),y(0)=1,y(0)=1

16. y+y=cos2t+2δ(tπ/2)3δ(tπ),y(0)=0,y(0)=1

17. yy=4et5δ(t1)+3δ(t2),y(0)=0,y(0)=0

18. y+2y+y=etδ(t1)+2δ(t2),y(0)=0,y(0)=1

19. y+y=f(t)+δ(t2π),y(0)=0,y(0)=1,

f(t)={sin2t,0t<π,0,tπ.

20. y+4y=f(t)+δ(tπ)3δ(t3π/2),y(0)=1,y(0)=1,

f(t)={1,0t<π/2,2,tπ/2

Q8.7.2

21. y+y=δ(t),y(0)=1,y(0)=2

22. y4y=3δ(t),y(0)=1,y(0)=7

23. y+3y+2y=5δ(t),y(0)=0,y(0)=0

24. y+4y+4y=δ(t),y(0)=1,y(0)=5

25. 4y+4y+y=3δ(t),y(0)=1,y(0)=6

Q8.7.3

In Exercises 8.7.26-8.7.28, solve the initial value problem ayh+byh+cyh={0,0t<t01/h,t0t<t0+h,0,tt0+h,yh(0)=0,yh(0)=0 where t0>0 and h>0. Then find w=L1(1as2+bs+c) and verify Theorem 8.7.1 by graphing w and yh on the same axes, for small positive values of h.

26. y+2y+2y=fh(t),y(0)=0,y(0)=0

27. y+2y+y=fh(t),y(0)=0,y(0)=0

28. y+3y+2y=fh(t),y(0)=0,y(0)=0

Q8.7.4

29. Recall from Section 6.2 that the displacement of an object of mass m in a spring–mass system in free damped oscillation is

my+cy+ky=0,y(0)=y0,y(0)=v0,

and that y can be written as

y=Rect/2mcos(ω1tϕ)

if the motion is underdamped. Suppose y(τ)=0. Find the impulse that would have to be applied to the object at t=τ to put it in equilibrium.

30. Solve the initial value problem. Find a formula that does not involve step functions and represents y on each subinterval of [0,) on which the forcing function is zero.

  1. yy=k=1δ(tk),y(0)=0,y(0)=1
  2. y+y=k=1δ(t2kπ),y(0)=0,y(0)=1
  3. y3y+2y=k=1δ(tk),y(0)=0,y(0)=1
  4. y+y=k=1δ(tkπ),y(0)=0,y(0)=0

This page titled 8.7E: Constant Coefficient Equations with Impulses (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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