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# Lectures 8+9: Exercises

### Suggestions for further reading

- Laplace transformation: Arfken (chaps. 15.8-15.12)
- Differential equations: Lyons (chap. 5)

### Questions for Review

- How is the Laplace transform defined? How does it differ from
the Fourier transform?
- Under which condition does the Laplace transform exist?
- Remember the following properties of the Laplace transformation:
- linearity
- convolution theorem
- Laplace transform of a derivative
- Laplace transform of an integral
- Laplace transform of a translated function f(t-b)
- Laplace transform of a function with rescaled argument f(at)
- ``damping'' theorem
- ``multiplication'' theorem
- Laplace transform of f(t)/t

- What is the Laplace transform of the delta function? What is
the Laplace transform of 1?
- What is a differential equation? What is the basic problem?
- Explain the following terminology:
- ordinary/partial DE
- DE of order n
- implicit/explicit representation
- system of k coupled DEs

- Can an n-th order DE be mapped onto first order DEs?
- Is the solution of a DE unique? What can you say about the
general solution of an n-th order DE?
- Explain:
- general/special solution
- initial/boundary conditions

- Explain the following special types of first order DEs
and how they can be solved:
- DE with separable variables
- homogeneous DE
- exact DE

- What is an integrating factor? How can it be used to solve
arbitrary first order linear DEs?

### Problems