Final Exam Review Topics

• Systems of Equations
  • Principle of Superposition
  • Wronskian for solutions to a system of D.E.'s
  • Homogeneous linear systems with constant coefficients
    • distinct, real eigenvalues
    • complex eigenvalues
    • repeated eigenvalues
  • Behavior of solutions as $t \to \infty$
  • Converting an nth order D.E. to a system of first order equations
• Partial Differential Equations
  • Fourier Series
    • computing Fourier coefficients
    • Fourier Convergence Theorem
      • conditions for convergence
      • convergence at points of discontinuity
  • The technique of separation of variables
  • Inner product of functions
    • orthogonal functions
    • orthogonality relations for sine and cosine
  • Even/Odd functions
    • adding and multiplying even and odd functions
    • integral properties of even and odd functions
    • Even/Odd periodic extensions
    • Fourier Sine Series
    • Fourier Cosine Series
    • f(0) = 0 for odd functions
      • Also need f(L) = 0 = f(-L) for odd functions if the Fourier Series is to converge everywhere to that function.
  • Heat equation
    • general solution for homogeneous boundary conditions
    • solving for arbitrary constants using the initial condition
    • modified procedure for non-homogeneous boundary conditions
  • Wave equation
    • general solution
    • solving for arbitrary constants using the initial conditions

Important Theorems:

• Theorem 7.4.1 (Principle of Superposition for solutions to systems of equations)
• Theorem 7.4.3 (significance of Wronskian for systems of equations)
• Theorem 10.3.1 (Fourier Convergence Theorem)

New terms and definitions you should know:

1.

eigenvalue (for boundary value problems)

2.

eigenfunction

3.

Hermitian matrix/system

4.

periodic function

5.

fundamental period

6.

orthogonal functions

7.

mutually orthogonal functions

8.

piecewise continuous

9.

even/odd function

Frequently Asked Questions

1.

What sorts of graphing will we need to do on the exam?
You should know how to plot periodic functions by hand. I will not ask you to do other stuff like plotting solutions to heat and wave equations, plotting phase portraits, etc.

2.

Do we have to memorize the solutions to the heat equation and wave equation?
Yes.

3.

Do we need to know how to derive them?
You should understand the technique of separation of variables as we apply it to solving partial differential equations. However, I will not ask you to derive the solution to a partial differential equation from scratch.

4.

Will we have to work heat equation problems for bars with insulated ends?
No.

5.

Is Laplace's equation on the exam?
No.