Exam 1: Friday, Oct. 1 -- Room 303 SH

Review Class - in class on Wed. Sept 29

Tentative Topics - all material covered in class unless specifically omitted

1. Chapter 5 - series solutions
   • 5.1 - Review of power series - ratio test; radius of convergence; analytic function
   • 5.2-5.3 - Solution near ordinary pt.; Thm 5.3.1; construct general solution; recurrence relation
   • 5.5 - Euler's equation - construct solution; Thm 5.5.1
   • 5.4,5.6-5.7 - Solution near regular singular point; method of Frobenius; Thm 5.7.1; indicial equation; exponents
   • Bessel's equation - recognize and find general solution.
2. Chapter 7 - Linear systems
   • 7.1 - convert 2nd order ODE to system; solve by elimination; terminology
   • 7.2-7.3 - linear algebra review; solve Ax=b; linear independence; eigenvalue problem; Hermitian
   • 7.4 - Theory of linear systems - superposition; Wronskian
   • 7.5-7.6 Solving homogeneous linear systems; equilibrium solution; phase plane; stability; complex eigenvalues
   • 7.7 - Fundamental matrix and matrix exponential
   • 7.8 - repeated root case; generalized eigenvector; improper node.
   • 7.9 - Nonhomogeneous eqn; variation of parameters only.

Exam 2: Monday, November 22 - Room: 303 SH

Review Class - in class on Fri. Nov. 19

Tentative Topics - all material covered in class unless specifically omitted

1. Chapter 7 - Review of matrices; solving linear equations; eigenvalue problem; homogeneous linear system of ODEs with constant coefficients - all cases of eigenvalues; phase portraits.
2. Chapter 9
   • 9.1 - Review of phase plane; linear systems
   • 9.2 - autonomous systems; types of stability; pendulum problem.
   • 9.3 - almost linear systems; conditions on system; results in table 9.3.1.
   • 9.4-9.5 - examples of complete analysis; look over predator-prey model.
   • 9.6 - Understand the energy function version of Liaponov stability.
   • 9.7 - periodic solutions; limit cycles; convert to polar coordinates; Using Theorems 9.7.1-9.7.3.
   • 9.8 - OMIT
3. Chapter 10 -
   • 10.1 - Two-point BVP; equilibrium solution of heat equation; eigenvalue-BVP
   • 10.2-10.4 - Fourier series
   • 10.5 - SOV applied to heat equation - know all steps
   • 10.6 - Heat equation with nonhomogeneous bc's; other types of bc's - insulated ends, etc.
4. Chapter 11
   • 11.1 - What is Sturm-Lioville BVP; find i eigenvalues and eigenfunctions.
   • 11.2 - Lagrange's identity -- What is it? How is it used? inner product; Key results for S-L BVP; self-adjoint.
   • 11.3 - Nonhomogeneous S-L BVP; construct solution by eigenfunction expansion; Fredholm Alternative Thm.

Final Exam Discussion