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 y) = 2t, e-2t y = t2 + c, c = 2, therefore y = (t2
Separable dierential equations (Sect. 1.3).
Separable ODE.
Solutions to separable ODE.
Explicit and implicit solutions.
Euler homogeneous equations.
Separable ODE.
Denition
A separable dierential equation on the function y has the form
h(y ) y (t ) = g (t
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 Wronskian of two functions.
General and fundamental solutions
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 dierential equations.
Theorem (Variable coecients)
Given con
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 for particular situations.
Radioactive decay
Remarks:
(a
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.
Review: Linear constant coecient equations
Denition
Gi
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 order ODE on the unknown
function y : R R is called se
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.
Denition
Given an open rectangle R = (t1 , t2 ) (u1 , u
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 dependent and independent functions.
General and fundamental so
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 of dierential equations.
Denition
A dierential equatio
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 of dierential equations.
Denition
A dierential equatio
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: Linear Dierential Equations
Theorem (Variable coecients)
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 particular situations.
Radioactive decay
Remarks:
(a) Ra
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.
Denition
The dierential equation for the unknown functi
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.
Review: Linear constant coecient equations
Denition
Gi
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 equations.
Review: Second order linear ODE.
Denition
Gi
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 numbers.
A real-valued fundamental and general solutions.
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 coecients equations.
Variable coecients equations.
Review
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 Exam 2 Review: http:/math.msu.edu/mlc
Exam Covers:
Varia
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(y ) = f
The General Solution Theorem
The Undetermined C
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 Equations
Characteristic Polynomial with Complex Roots
Second O
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 Wronskian
Second Order Linear Equations ( 2.1)
Review: T
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
General and Fundamental Solutions
Second Order Linear Equatio
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)
Review: Linear Dierential Equations
The Picard-Lindelf Th
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 handout.
No notes, no books, no calculators, no phones.
Linear Constant Coecients Equations ( 1.1)
Overview of Dierential Equations
Linear Ordinary Dierential Equations
Integrating a Dierential Equation
Linear Constant Coecients Equations
The Integrating Factor Method
The Initial Value Problem
Linear Constant
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
Generalization of the Integrating Factor Method
Exact Equations (
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: A Nonlinear Equation
Linear Variable Coecient Equation