4.8E: Undetermined Coefficients for Higher Order Equations (Exercises)

Q4.8.1

In Exercises 4.8.1-4.8.9 find the general solution.

1. \(y'''-y''-y'+y=e^{2x}(10+3x)\)

2. \(y'''+y''-2y=-e^{3x}(9+67x+17x^2)\)

3. \(y'''-6y''+11y'-6y=e^{2x}(5-4x-3x^2)\)

4. \(y'''+2y''+y'=-2e^{-x}(7-18x+6x^2)\)

5. \(y'''-3y''+3y'-y=e^x(1+x)\)

6. \(y^{(4)}-2y''+y=-e^{-x}(4-9x+3x^2)\)

7. \(y'''+2y''-y'-2y=e^{-2x}\left[(23-2x)\cos x+(8-9x)\sin x\right]\)

8. \(y^{(4)}-3y'''+4y''-2y'=e^x\left[(28+6x)\cos 2x+(11-12x)\sin2x\right]\)

9. \(y^{(4)}-4y'''+14y''-20y'+25y=e^x\left[(2+6x)\cos 2x+3\sin2x\right]\)

10. \(y'''+3y''+3y'+y=12e^{-x} +9\cos 2x-13\sin2x\)

Q4.8.2

In Exercises 4.8.10-4.8.15 solve the initial value problem and graph the solution.

11. \(y'''-2y''-5y'+6y=2e^x(1-6x),\quad y(0)=2, \quad y'(0)=7,\quad y''(0)=9\)

12. \(y'''-y''-y'+y=-e^{-x}(4-8x),\quad y(0)=2, \quad y'(0)=0,\quad y''(0)=0\)

13. \(4y'''-3y'-y=e^{-x/2}(2-3x),\quad y(0)=-1, \quad y'(0)=15,\quad y''(0)=-17\)

14. \(y^{(4)}+2y'''+2y''+2y'+y=e^{-x}(20-12x),\, y(0)=3,\; y'(0)=-4,\; y''(0)=7,\; y'''(0)=-22\)

15. \(y'''+2y''+y'+2y=30\cos x-10\sin x, \quad y(0)=3,\quad y'(0)=-4,\quad y''(0)=16\)

16. \(y^{(4)}-3y'''+5y''-2y'=-2e^x(\cos x-\sin x),\; y(0)=2,\; y'(0)=0,\; y''(0)~=~-1, \; y'''(0)=-5\)