Ordinary Differential Equations
Final Review
Shurong Sun
University of Jinan
Semester 1, 2011-2012
1. Introduction
(1). Definitions
(a) Differential Equation, Mathematical Model
(b) Order, Linear, Nonlinear
(c) Solution, General Solution, Particular Solution
(d) Initial Condition, Initial Value Problem (IVP)
(e) Direction (Slope) Field, Integral curves
(2). Important Skills
(a) Derive differential equations that mathematically model
simple problems.
(b) Know what an initial value problem is, and how to show a
given function is a solution to one.
(c) Know how to classify differential equations using order, and
linearity.
2. First Order Differential Equations
(1). Definitions
(a) Separable differential equations
(b) Homogeneous differential equations
(c) Exact equations, Integrating factors
(d) Linear first order differential equation，variation of constants
(e) Bernoulli Equations
(f) Equations with linear coefficients
(g) Lipschitz condition
(h) local Lipschitz condition
(i) Existence and Uniqueness of Solutions to initial value problems
(j) The method of successive approximations, or Picard iteration
(k) Continuation of solutions, Saturated solutions, Maximal interval
of existence
(l) Continuous dependence on initial conditions and parameters
(m) Smooth dependence on initial conditions
(2). Theorems
(a) Existence and uniqueness of solutions to first order IVP’s
(b) Existence and uniqueness of solutions to linear first order ODE’s
(c) Existence of saturated solutions
(d) Classification of saturated solutions
(e) Continuous dependence on initial conditions and parameters
(f) Smooth dependence on initial conditions
(3). Important Skills
(a) If the differential equation is separable, then solve it by method of
Separation of Variables.
(b) If it isn’t separable, is it homogenous or equations with linear
coefficients, then solve it using substitutions that would convert it
into a separable equation.
(c) If the differential equation is linear, solve it by variation of constants ,
or compute the integrating factor and then the general solution.
(d) If it isn’t linear, is it Bernoulli equations, then solve it using
substitutions that would convert it into a linear one.
(e) If the differential equation is not separable or linear, is it exact? If so,
solve it by finding the primitive function.
(f) If the differential equation is not exact, compute the integrating factor,
and then convert it into a exact one.
(g) Understand the three steps in the process of mathematical
modeling: construction of the model, analysis of the model,
and comparison with experiment or observation.
(h) Understand Relevant Applications: Mixing Problems, etc.
(i) Determine the existence and uniqueness of solutions to
differential equations and compute the existence interval.
(j) Be able to construct the Picard iterates and Compute the
approximate solution for the initial-value problem
(k) Determine the maximal interval of saturated solutions
3. Higher Order Linear Equations
(1). Definitions:
(a) Linear vs. nonlinear
(b) Homogeneous vs. nonhomogeneous
(c) Principle of superposition
(d) Linear independence and dependence, Wronskian
(e) General Solution, Particular Solution
(f) Fundamental Set of Solutions
(g) Characteristic Polynomial, Characteristic Equation
(h) Method of Undetermined Coefficients
(i) Method of Variation of Parameters
(j) Cauchy-Euler equation
(2). Theorems
(a) Existence and uniqueness
(b) Principle of superposition.
(c) Properties of Wronskian
(d) Abel’s formula
(e) Representing general solutions to higher order linear
homogeneous ODE’s
(f) General solutions to linear nonhomogeneous ODE’s.
(g) Representing general solutions to second order linear
homogeneous ODE’s
(3). Important Skills
(a) Determine the dependence of a set of functions
(b) Be able to write down the characteristic equation to a
homogeneous equation with constant coefficients, and compute
characteristic roots and determine its type: real and distinct,
repeated, or complex conjugate.
(c) Be able to write down fundamental solution sets to homogeneous
equations.
(d) To find particular solutions by the Method of Undetermined
coefficients or Variation of Parameters.
(e) Be able to the Laplace transform method
(f) Be able to the power series method
(g) Be able to the method of Reduction of Order
(h) Be able to write down the index equation to a Cauchy-Euler
equation, and compute its roots and determine its type: real
and distinct, repeated, or complex conjugate.
(i) Be able to write down fundamental solution sets to Cauchy-Euler
equation
(j) Understand Relevant Applications: Vibrating Springs, etc.
4. System of first order linear differential equations
(1). Definitions:
(a) Linear vs. nonlinear
(b) Homogeneous vs. nonhomogeneous
(c) Principle of superposition
(d) Linear independence and dependence, Wronskian
(e) General Solution, Particular Solution
(f) Fundamental Set of Solutions, Fundamental matrix
(g) Characteristic Polynomial, Characteristic Equation,
eigenvector, generalized eigenvector
(h) matrix exponential function
(i) Method of Undetermined Coefficients
(j) Method of Variation of Parameters
(2). Theorems
(a) Existence and uniqueness
(b) Principle of superposition.
(c) Properties of Wronskian
(d) Abel’s formula
(e) Representing general solutions to higher order linear
homogeneous ODE’s
(f) General solutions to linear nonhomogeneous ODE’s.
(g) Representing general solutions to second order linear
homogeneous ODE’s
(3). Important Skills
(a) Convert higher differential equations into a system of first-order
equations.
(b) Determine the dependence of a set of functions
(c) Be able to write down the characteristic equation to a
matrix, compute eigenvalues and eigenvector, generalized
eigenvector corresponding to each eigenvalue
(d) Be able to write down solutions to each eigenvalue and eigenvector,
generalized eigenvector corresponding to each eigenvalue
(e) Be able to write down fundamental solution sets and Fundamental
matrix to homogeneous equations.
(f) To find particular solutions by the Method of Undetermined
coefficients or Variation of Parameters.
(g) Be able to the Laplace transform method
(h) Understand Relevant Applications: Mixing Problems, etc.