Study guide for math251, Ordinary and Partial differential equations

This is no exclusive list for studying for the final, but a more or less concise summary about what you should take home from the course.

1.Introduction

1.3	You should understand the difference between an ODE and a PDE, linear and nonlinear. For the lin. case homogeneous and inhomog.

2. First Order Differential Equations

2.1	Method of integrating factor
2.2	I'm sure you know this
2.3	How to abstract a first order ODE and initial values from a text problem - no mixing problems, but the others might be (see examples in the book)
2.4	Existence and uniqueness of solutions of lin. ODEs
2.5	autonomous equations, critical points, and stability analysis

3. Second Order Linear Equations

3.1, 3.2	solving homogeneous equations with constant coefficients Existence and Uniqueness of solutions of lin. ODE.
3.3	linear independence, Wronskian, Abel's theorem - reduction of order is not on the final
3.4	complex roots of the characteristic equation, sin/ cos solutions
3.5	repeated roots, second solution
3.6	easy inhomogenities, ansatz with undetermined coefficients
3.7	general inhomogenities, Variation of parameters (maybe the formula)
3.8	damped harmonic oscillator, 3 cases of damping tricks like unifying 2 harmonic functions into one (at least the amplitude)
3.9	- not on the final

6. Laplace Transform

6.1	Identifying a Laplace transform and the transformed function
6.2	solving ODEs via Laplace transform Partial fraction decomposition There will be a table of Laplace transforms similar to the one in the book
6.3	expressing piecewise defined functions and how to transform back e^-csF(s) (see also table). How to transform back F(s-c) (see table).
6.4	Transforming 6.3
6.5	Laplace transform of Dirac's delta function δ(t-c).

7. Systems of first order linear differential equations

7.1, 7.2, 7.3	solving lin eqns 2x2, inverting 2x2 matrices, determining eigenvectors, eigenvalues
7.4	existence and uniqueness of solutions of lin. ODEs
7.5	general solution via eigenvalues, eigenvectors
7.6	complex eigenvalues and general real solution
7.7	how to compute the fundamental matrix Φ(t) (from the general solution).
7.8	repeated eigenvalues, generalized eigenvector/ complementary vector; general solution of the ODEs
9.1	Type and stability of (0,0) for linear systems tricks like restrictions of initial data for converging solutions

9. Nonlinear differential equations and stability

9.2	autonomous systems, their critical points, linearization, type and stability pendulum formulation and translation into a 1st order ODEs
9.5	- will not be on the exam

10. Partial Differential Equations and Fourier Series (60% of the final)

10.1	2nd order ODE boundary value problem, the solution of the particular one we had.
10.2	Fourier series, periodicity and formula for Fourier coefficients
10.3	Convergence of the Fourier series for piecewise cont. functions with piecewise cont. deriv.
10.4	definition and simplification in computing the Fourier coefficients
10.5	method of separation of variables and their homogeneous boundary conditions; optional: the solution of the heat conduction eqn with homogeneous Dirichlet bdy-values ( u(t,0)=0=u(t,L) ).
10.6	steady state solution and treatment of non-homogeneous bdy-conditions ( u(t,x)=u_∞(x)+ 10.5 ).
10.7	separation of variables and the homog. bdy-condit. understanding how the solution should look like ( sin(πnx/L)cos(πnct/L) +…sin(πnct/L) ). How to formulate initial conditions for this PDE
10.8	separation of variables and the homog. bdy-condit. How the solutions should look like ( sin(πnx/a)sinh(πny/a) ).