5.3.1: Constant Coefficient Homogeneous Equations (Exercises)

In Exercises 1-12 find the general solution.

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

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

3. \(y''+8y'+7y=0\)

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

5. \(y'' +2y'+10y=0\)

6. \(y''+6y'+10y=0\)

7. \(y''-8y'+16y=0\)

8. \(y''+y'=0\)

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

10. \(y''+6y'+13y=0\)

11. \(4y''+4y'+10y=0\)

12. \(10y''-3y'-y=0\)

In Exercises 13-21 solve the initial value problem.

13. \(y''+14y'+50y=0, \quad y(0)=2,\quad y'(0)=-17\)

14. \(6y''-y'-y=0, \quad y(0)=10,\quad y'(0)=0\)

15. \(6y''+y'-y=0, \quad y(0)=-1,\quad y'(0)=3\)

16. \(4y''-4y'-3y=0, \quad y(0)={13\over 12},\quad y'(0)={23 \over 24}\)

17. \(4y''-12y'+9y=0, \quad y(0)=3,\quad y'(0)={5\over 2}\)

18. \(y''+7y'+12y=0, \quad y(0)=-1,\quad y'(0)=0\)

19. \(y''-6y'+9y=0, \quad y(0)=0,\quad y'(0)=2\)

20. \(36y''-12y'+y=0, \quad y(0)=3,\quad y'(0)={5\over2}\)

21. \(y''+4y'+10y=0, \quad y(0)=3,\quad y'(0)=-2\)

22. Prove: If the characteristic equation of

\[ay''+by'+cy=0 \tag{A}\]

has a repeated negative root or two roots with negative real parts, then every solution of (A) approaches zero as \(x\to\infty\).

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