4.4E: Higher Order Constant Coefficient Homogeneous Equations (Exercises)
- Page ID
- 43353
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Q4.4.1
In Exercises 4.4.1-4.4.14 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. \(y^{(4)}-4y'''+7y''-6y'+2y=0\)
Q4.4.2
In Exercises 4.4.15-4.4.27 solve the initial value problem. Graph the solution for Exercises 4.4.17-4.4.19 and 4.4.27.
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. \(4y^{(4)}+8y'''+19y''+32y'+12y=0, \quad y(0)=3,\quad y'(0)=-3,\quad y''(0)= -\frac{7}{2},\quad y'''(0)=\frac{31}{4}\)
Q4.4.3
28. Find a fundamental set of solutions of the given equation.
- \(y'''-y=0\)
- \(y'''+y=0\)
- \(y^{(4)}+64y=0\)
- \(y^{(6)}-y=0\)
- \(y^{(6)}+64y=0\)