Separable dierential equations (Sect. 2.2).
Separable ODE.
Solutions to separable ODE.
Explicit and implicit solutions.
Homogeneous equations.
Separable ODE.
Denition
Given functions h, g : R R, a rst order ODE on the unknown
function y : R R is called se
Non-homogeneous equations (Sect. 3.5).
We study: y + a1 y + a0 y = b (t ).
Operator notation and preliminary results.
Summary of the undetermined coecients method.
Using the method in few examples.
The guessing solution table.
Operator notation and prelim
Autonomous systems (Sect. 2.5).
Denition and examples.
Qualitative analysis of the solutions.
Equilibrium solutions and stability.
Population growth equation.
Denition and examples
Denition
A rst order ODE on the unknown function y : R R is called
autonom
Modeling with rst order equations (Sect. 2.3).
Main example: Salt in a water tank.
The experimental device.
The main equations.
Analysis of the mathematical model.
Predictions for particular situations.
Salt in a water tank.
Problem: Describe the salt con
Exact equations (Sect. 2.6).
Exact dierential equations.
The Poincar Lemma.
e
Implicit solutions and the potential function.
Generalization: The integrating factor method.
Exact dierential equations.
Denition
Given an open rectangle R = (t1 , t2 ) (u1 , u
On linear and non-linear equations.(Sect. 2.4).
Review: Linear dierential equations.
Non-linear dierential equations.
Properties of solutions to non-linear ODE.
The Bernoulli equation.
Review: Linear dierential equations.
Theorem (Variable coecients)
Give
Review 2 for Exam 1.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers:
Linear equations (2.1).
Separable equations (2.2).
Homogeneous equations (2.2).
Modeling using dierential e
Second order linear ODE (Sect. 3.1).
Second order linear dierential equations.
Superposition property.
Constant coecients equations.
The characteristic equation.
The main result.
Second order linear dierential equations.
Denition
Given functions a1 , a0 ,
Variable coecients second order linear ODE (Sect. 3.2).
Summary: The study the main properties of solutions to second
order, linear, variable coecients, ODE.
Review: Second order linear ODE.
Existence and uniqueness of solutions.
Linearly dependent and in
The integrating factor method (Sect. 2.1)
Overview of dierential equations.
Linear Ordinary Dierential Equations.
The integrating factor method.
Constant coecients.
The Initial Value Problem.
Variable coecients.
Read:
The direction eld. Example 2 in Secti
Second order linear homogeneous ODE (Sect. 3.3).
Review: On solutions of y + a1 y + a0 y = 0.
Characteristic polynomial with complex roots.
Two main sets of fundamental solutions.
A real-valued fundamental and general solutions.
Application: The RLC circu
Second order linear homogeneous ODE (Sect. 3.4).
Review: On solutions of y + a1 y + a0 y = 0.
Repeated roots as a limit case.
Main result for repeated roots.
Reduction of the order method:
Constant coecients equations.
Variable coecients equations.
Review
Non-homogeneous equations (Sect. 3.6).
We study: y + p (t ) y + q (t ) y = f (t ).
Method of variation of parameters.
Using the method in an example.
The proof of the variation of parameter method.
Using the method in another example.
Method of variation