Math 235
section 202
HW #1 Solution
1 dy - y = te2t , y(0) = 2 2 dt
1. (3 pts) Find the solution of the initial value problem: y - 2y = 2te2t , (t) = e e-2t y - 2e-2t y = 2t,
-2dt
= e-2t ,
(e-2t
Variable coecients second order linear ODE (Sect. 2.1).
Second order linear ODE.
Superposition property.
Existence and uniqueness of solutions.
Linearly dependent and independent functions.
The Wronsk
On linear and non-linear equations. (Sect. 1.6).
Review: Linear dierential equations.
Non-linear dierential equations.
Properties of solutions to non-linear ODE.
Direction Fields.
Review: Linear diere
Modeling with rst order equations (Sect. 1.5).
Radioactive decay.
Carbon-14 dating.
Salt in a water tank.
The experimental device.
The main equations.
Analysis of the mathematical model.
Predictions f
Linear Variable coecient equations (Sect. 2.1)
Review: Linear constant coecient equations.
The Initial Value Problem.
Linear variable coecients equations.
The Bernoulli equation: A nonlinear equation.
Separable dierential equations (Sect. 1.3).
Separable ODE.
Solutions to separable ODE.
Explicit and implicit solutions.
Homogeneous equations.
Separable ODE.
Denition
Given functions h, g : R R, a rst
Exact equations (Sect. 1.4).
Exact dierential equations.
The Poincar Lemma.
e
Implicit solutions and the potential function.
Generalization: The integrating factor method.
Exact dierential equations.
Variable coecients second order linear ODE (Sect. 2.1).
Second order linear ODE.
Existence and uniqueness of solutions.
Operator notation.
Linear operator and Superposition property.
Linearly dependen
The integrating factor method (Sect. 1.1)
Overview of dierential equations.
Linear Ordinary Dierential Equations.
The integrating factor method.
Constant coecients.
The Initial Value Problem.
Overview
The integrating factor method (Sect. 1.1)
Overview of dierential equations.
Linear Ordinary Dierential Equations.
The integrating factor method.
Constant coecients.
The Initial Value Problem.
Overview
On linear and Non-inear Equations (Sect. 1.6)
Review: Linear Dierential Equations
The Picard-Lindelf Theorem
o
The Picard Iteration
Properties of Solutions to Non-Linear ODE
Direction Fields
Review: L
Modeling with rst order equations (Sect. 1.5).
Radioactive decay.
Carbon-14 dating.
Salt in a water tank.
The experimental device.
The main equations.
The equation for the salt mass.
Predictions for p
Exact equations (Sect. 1.4).
Exact dierential equations.
The Poincar Lemma.
e
Implicit solutions and the potential function.
Generalization: The integrating factor method.
Exact dierential equations.
Linear Variable coecient equations (Sect. 1.2)
Review: Linear constant coecient equations.
The Initial Value Problem.
Linear variable coecients equations.
The Bernoulli equation: A nonlinear equation.
Second order linear ODE (Sect. 2.2).
Review: Second order linear dierential equations.
Idea: Soving constant coecients equations.
The characteristic equation.
Solution formulas for constant coecients
Second order linear homogeneous ODE (Sect. 2.3).
Review: On solutions of y + a1 y + a0 y = 0.
Characteristic polynomial with complex roots.
Two main sets of fundamental solutions.
Review of Complex nu
Second order linear homogeneous ODE (Sect. 2.4).
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 co
Review for Exam 2.
6 Problems, 55 Minutes, in Recitation Rooms
100 Grading Attempts
Problems Similar to Homeworks
Integration Table Provided in Handout
No Notes, No Books, No Calculators
MLC MTH 235 E
Nonhomogeneous Equations ( 2.4)
The Problem: L(y ) = f
The General Solution Theorem
The Undetermined Coecients Method
The Variation of Parameters Method
Nonhomogeneous Equations ( 2.4)
The Problem: L(
Second Order Linear Equations ( 2.3)
Review: Second Order Linear Dierential Equations
Idea: Solving Constant Coecients Equations
The Characteristic Equation
Main Result for Constant Coecients Equation
Second Order Linear Equations ( 2.1)
Review: The General Solution Theorem
The Wronskian of Two Functions
The Wronskian an Linear Dependence
The Wronskian and Linear Independence
Abels Theorem on the W
Second Order Linear Equations ( 2.1)
Second Order Linear Dierential Equations
Solutions to the Initial Value Problem
Linear Operators and the Superposition Property
Linearly Dependent Functions
Genera
Nonlinear Equations ( 1.6)
Review: Linear Dierential Equations
The Picard-Lindelf Theorem
o
The Picard Iteration
Properties of Solutions to Nonlinear ODE
Direction Fields
Nonlinear Equations ( 1.6)
Re
Review for Exam 1
On Webwork Exam Server, in Recitation Rooms and Times.
100 grading attempts.
6 problems, 55 minutes.
Problems similar to webwork homework problems.
Integration table provided in the
Linear Constant Coecients Equations ( 1.1)
Overview of Dierential Equations
Linear Ordinary Dierential Equations
Integrating a Dierential Equation
Linear Constant Coecients Equations
The Integrating F
Exact Equations ( 1.4)
Review: Exact Dierential Equations
Nonexact Equations
Example of a Nonexact Equation
Linear Equations are Nonexact
The Integrating Factor Method for Linear Equations
Generalizat
Linear Variable Coecient Equations ( 1.2)
Review: Linear Constant Coecient Equations
Linear Variable Coecients Equations
The Integrating Factor Method
The Initial Value Problem
The Bernoulli Equation: