In Problems 1–20, use either the Laplace transform or a Fourier transform to solve the given boundary-value problem. Make assumptions about boundedness where necessary. The symbols are constants.

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
    [Hint: Use the identity sinh(x − y) = sinh x cosh y − cosh x sinh y, and then use Problem 8 in Exercises 15.1.]
13. 
14. 
15. Show that a solution of the BVP

    

    is

16. 
17. 
18. 
19. 
20. 
21. Solve the boundary-value problem

    

    using the Laplace transform. Give two different forms of the solution

22. 1. Solve the BVP in Problem 21 using a Fourier transform.
    2. Use a CAS to carry out an integration to show that the answer in part (a) is equivalent to one of the answers in Problem 21.

In Problems 23–26, use a finite Fourier transform to solve the given boundary-value problem. The symbols are constants.