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8.5E: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Q8.5.1

In Exercises 8.5.1-8.5.20 use the Laplace transform to solve the initial value problem. Graph the solution for Exercise 8.5.6, 8.5.9, 8.5.13, and 8.5.19.

1. y+y={3,0t<π,0,tπ,y(0)=0,y(0)=0

2. y+y={3,0t<4,;2t5,t>4,y(0)=1,y(0)=0

3. y2y={4,0t<1,6,t1,y(0)=6,y(0)=1

4. yy={e2t,0t<2,1,t2,y(0)=3,y(0)=1

5. y3y+2y={0,0t<1,1,1t<2,1,t2,y(0)=3,y(0)=1

6. y+4y={|sint|,0t<2π,0,t2π,y(0)=3,y(0)=1

7. y5y+4y={1,0t<11,1t<2,0,t2,y(0)=3,y(0)=5

8. y+9y={cost,0t<3π2,sint,t3π2,y(0)=0,y(0)=0

9. y+4y={t,0t<π2,π,tπ2,y(0)=0,y(0)=0

10. y+y={t,0t<π,t,tπ,y(0)=0,y(0)=0

11. y3y+2y={0,0t<2,2t4,t2,,y(0)=0,y(0)=0

12. y+y={t,0t<2π,2t,t2π,y(0)=1,y(0)=2

13. y+3y+2y={1,0t<2,1,t2,y(0)=0,y(0)=0

14. y4y+3y={1,0t<1,1,t1,y(0)=0,y(0)=0

15. y+2y+y={et,0t<1,et1,t1,y(0)=3,y(0)=1

16. y+2y+y={4et,0t<1,0,t1,y(0)=0,y(0)=0

17. y+3y+2y={et,0t<1,0,t1,y(0)=1,y(0)=1

18. y4y+4y={e2t,0t<2,e2t,t2,y(0)=0,y(0)=1

19. y={t2,0t<1,t,1t<2,t+1,t2,y(0)=1,y(0)=0

20. y+2y+2y={1,0t<2π,t,2πt<3π,1,t3π,y(0)=2,y(0)=1

Q8.5.2

21. Solve the initial value problemy=f(t),y(0)=0,y(0)=0,wheref(t)=m+1,mt<m+1,m=0,1,2,.

22. Solve the given initial value problem and find a formula that does not involve step functions and represents y on each interval of continuity of f.

  1. y+y=f(t),y(0)=0,y(0)=0;
    f(t)=m+1,mπt<(m+1)π,m=0,1,2,.
  2. y+y=f(t),y(0)=0,y(0)=0;
    f(t)=(m+1)t,2mπt<2(m+1)π,m=0,1,2, HINT: You'll need the formula 1+2++m=m(m+1)2.
  3. y+y=f(t),y(0)=0,y(0)=0;
    f(t)=(1)m,mπt<(m+1)π,m=0,1,2,.
  4. yy=f(t),y(0)=0,y(0)=0;
    f(t)=m+1,mt<(m+1),m=0,1,2,.
    HINT: You will need the formula 1+r+...+rm=1rm+11r(r1).
  5. y+2y+2y=f(t),y(0)=0,y(0)=0;
    f(t)=(m+1)(sint+2cost),2mπt<2(m+1)π,m=0,1,2,.
    (See the hint in d.)
  6. y3y+2y=f(t),y(0)=0,y(0)=0;
  7. f(t)=m+1,mt<m+1,m=0,1,2,.
    (See the hints in b and d.)

23.

  1. Let g be continuous on (α,β) and differentiable on the (α,t0) and (t0,β). Suppose A=limtt0g(t) and B=\lim_{t\to t_0+}g'(t) both exist. Use the mean value theorem to show that \lim_{t\to t_0-}{g(t)-g(t_0)\over t-t_0}=A\quad\mbox{ and }\quad \lim_{t\to t_0+}{g(t)-g(t_0)\over t-t_0}=B.\nonumber
  2. Conclude from (a) that g'(t_0) exists and g' is continuous at t_0 if A=B.
  3. Conclude from (a) that if g is differentiable on (\alpha,\beta) then g' can’t have a jump discontinuity on (\alpha,\beta).

24.

  1. Let a, b, and c be constants, with a\ne0. Let f be piecewise continuous on an interval (\alpha,\beta), with a single jump discontinuity at a point t_0 in (\alpha,\beta). Suppose y and y' are continuous on (\alpha,\beta) and y'' on (\alpha,t_0) and (t_0,\beta). Suppose also that ay''+by'+cy=f(t) \tag{A} on (\alpha,t_0) and (t_0,\beta). Show that y''(t_0+)-y''(t_0-)={f(t_0+)-f(t_0-)\over a}\ne0.\nonumber
  2. Use (a) and Exercise 8.5.23c to show that (A) does not have solutions on any interval (\alpha,\beta) that contains a jump discontinuity of f.

25. Suppose P_0,P_1, and P_2 are continuous and P_0 has no zeros on an open interval (a,b), and that F has a jump discontinuity at a point t_0 in (a,b). Show that the differential equation P_0(t)y''+P_1(t)y'+P_2(t)y=F(t)\nonumber has no solutions on (a,b). HINT: Generalize the result of Exercise 8.5.24 and use Exercise 8.5.23c.

26. Let 0=t_0<t_1<\cdots <t_n. Suppose f_m is continuous on [t_m,\infty) for m=1,\dots,n. Let f(t)= \left\{\begin{array}{cl} f_m(t),&t_m\le t< t_{m+1},\quad m=1,\dots,n-1,\\[4pt] f_n(t),&t\ge t_n. \end{array}\right.\nonumber Show that the solution of

ay''+by'+cy=f(t), \quad y(0)=k_0,\quad y'(0)=k_1,\nonumber

as defined following Theorem 8.5.1, is given by

y=\left\{\begin{array}{cl} z_0(t),&0\le t<t_1,\\[4pt] z_0(t)+ z_1(t),&t_1\le t<t_2,\\[4pt] &\vdots\\[4pt] z_0+\cdots+z_{n-1}(t),&t_{n-1}\le t<t_n,\\[4pt] z_0+\cdots+ z_n(t),&t\ge t_n, \end{array}\right.\nonumber

where z_0 is the solution of

az''+bz'+cz=f_0(t), \quad z(0)=k_0,\quad z'(0)=k_1\nonumber

and z_m is the solution of

az''+bz'+cz=f_m(t)-f_{m-1}(t), \quad z(t_m)=0,\quad z'(t_m)=0\nonumber

for m=1,\dots,n.


This page titled 8.5E: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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