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# Lectures 8+9: Exercises

• 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?