Differential Equations and Linear Algebra, 4th Edition

Section 5.1

1. 1. $y(x)=\frac{5}{2}{e}^x-\frac{5}{2}{e}^{-x}$

2. 2. $y(x)=2e^{3x}-3e^{-3x}$

3. 3. $y(x)=3\cos 2x+4\sin 2x$

4. 4. $y(x)=10\cos 5x-2\sin 5x$

5. 5. $y(x)=2e^x-e^{2x}$

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

7. 7. $y(x)=6-8e^{-x}$

8. 8. $y(x)=\frac{1}{3}(14-2e^{3x})$

9. 9. $y(x)=2e^{-x}+x{e}^{-x}$

10. 10. $y(x)=3e^{5x}-2{xe}^{5x}$

11. 11. $y(x)=5e^x\sin x$

12. 12. $y(x)=e^{-3x}(2\cos 2x+3\sin 2x)$

13. 13. $y(x)=5x-2x^2$

14. 14. $y(x)=3x^2-16/x^3$

15. 15. $y(x)=7x-5x\ln x$

16. 16. $y(x)=2\cos (\ln x)+3\sin (\ln x)$

17. 21. Linearly independent

18. 22. Linearly independent

19. 23. Linearly independent

20. 25. Linearly dependent

21. 25. Linearly independent

22. 26. Linearly independent

23. 28. $y(x)=1-2\cos x-\sin x$

24. 29. There is no contradiction because if the given differential equation is divided by $x^2$ to get the form in Eq. (8), then theresulting coefficient functions $p(x)=-4/x$ and $q(x)=6/x^2$ are not continuous at $x=0$.

25. 33. $y\left(x\right)=c_1{e}^x+c_2{e}^{}$