7.2.1: Higher Order Homogeneous Equations (Exercises)

Last updated
Save as PDF

Page ID: 103558

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In Exercises 1-14 find the general solution.

1. \(y'''-3y''+3y'-y=0\)

2. \(y^{(4)}+8y''-9y=0\)

3. \(y'''-y''+16y'-16y=0\)

4. \(2y'''+3y''-2y'-3y=0\)

5. \(y'''+5y''+9y'+5y=0\)

6. \(4y'''-8y''+5y'-y=0\)

7. \(27y'''+27y''+9y'+y=0\)

8. \(y^{(4)}+y''=0\)

9. \(y^{(4)}-16y=0\)

10. \(y^{(4)}+12y''+36y=0\)

11. \(16y^{(4)}-72y''+81y=0\)

12. \(6y^{(4)}+5y'''+7y''+5y'+y=0\)

13. \(4y^{(4)}+12y'''+3y''-13y'-6y=0\)

14. \(y^{(4)}-4y'''+7y''-6y'+2y=0\)

In Exercises 15-27 solve the initial value problem.

15. \(y'''-2y''+4y'-8y=0, \quad y(0)=2,\quad y'(0)=-2,\; y''(0)=0\)

16. \(y'''+3y''-y'-3y=0, \quad y(0)=0,\quad y'(0)=14,\quad y''(0)=-40\)

17. \(y'''-y''-y'+y=0, \quad y(0)=-2,\quad y'(0)=9,\quad y''(0)=4\)

18. \(y'''-2y'-4y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=22\)

19. \(3y'''-y''-7y'+5y=0, \quad y(0)= \frac{14}{5},\quad y'(0)=0,\quad y''(0)=10\)

20. \(y'''-6y''+12y'-8y=0, \quad y(0)=1,\quad y'(0)=-1,\quad y''(0)=-4\)

21. \(2y'''-11y''+12y'+9y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=13\)

22. \(8y'''-4y''-2y'+y=0, \quad y(0)=4,\quad y'(0)=-3,\quad y''(0)=-1\)

23. \(y^{(4)}-16y=0, \quad y(0)=2,\; y'(0)=2,\; y''(0)=-2,\; y'''(0)=0\)

24. \(y^{(4)}-6y'''+7y''+6y'-8y=0, \quad y(0)=-2,\quad y'(0)=-8,\quad y''(0)=-14,\quad y'''(0)=-62\)

25. \(4y^{(4)}-13y''+9y=0, \quad y(0)=1,\quad y'(0)=3,\quad y''(0)=1,\quad y'''(0)=3\)

26. \(y^{(4)}+2y'''-2y''-8y'-8y=0, \quad y(0)=5,\quad y'(0)=-2,\quad y''(0)=6,\quad y'''(0)=8\)

27. \(4y^{(4)}+8y'''+19y''+32y'+12y=0, \quad y(0)=3,\quad y'(0)=-3,\quad y''(0)= -\frac{7}{2},\quad y'''(0)=\frac{31}{4}\)

In Exercises 28-33 find a fundamental set of solutions of the given equation, and verify that it is a fundamental set by evaluating its Wronskian. Form the general solution.

28. \((D-1)^2(D-2)y=0\)

29. \((D^2+4)(D-3)y=0\)

30. \((D^2+2D+2)(D-1)y=0\)

31. \(D^3(D-1)y=0\)

32. \((D^2-1)(D^2+1)y=0\)

33. \((D^2-2D+2)(D^2+1)y=0\)

In Exercises 34-43 find a fundamental set of solutions and form the general solution.

34. \((D^2+6D+13)(D-2)^2D^3y=0\)

35. \((D-1)^2(2D-1)^3(D^2+1)y=0\)

36. \((D^2+9)^3D^2y=0\)

37. \((D-2)^3(D+1)^2Dy=0\)

38. \((D^2+1)(D^2+9)^2(D-2)y=0\)

39. \((D^4-16)^2y=0\)

40. \((4D^2+4D+9)^3y=0\)

41. \(D^3(D-2)^2(D^2+4)^2y=0\)

42. \((4D^2+1)^2(9D^2+4)^3y=0\)

In Exercises 43-48 find the general solution on \((0,\infty)\)

43. \(x^3y'''+5x^2y''+7xy'+8y=0\)

44. \(x^3y'''-6y=0\)

45. \(x^3y'''+xy'-y=0\)

46. \(xy^{(4)}+6y'''=0\)

47. \(x^4y^{(4)}+6x^3y'''+9x^2y''+3xy'+y=0\)

48. \(x^3y'''-3x^2y''+6xy'-6y=0\)