Undetermined Coefficients –– Superposition

Slides:

Advertisements

Similar presentations

Trig – Section 4 Graphing Sine and Cosine Objectives: To graph sine and cosine curves To find amplitude, period and phase shifts.

Advertisements

Boyce/DiPrima 9th ed, Ch 3.5: Nonhomogeneous Equations;Method of Undetermined Coefficients Elementary Differential Equations and Boundary Value Problems,

Chapter 2: Second-Order Differential Equations

Ch 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients

Rules for Differentiating Univariate Functions Given a univariate function that is both continuous and smooth throughout, it is possible to determine its.

CHAPTER Continuity CHAPTER Derivatives of Polynomials and Exponential Functions.

Basic Mechanical Engineering Courses

1 st order linear differential equation: P, Q – continuous. Algorithm: Find I(x), s.t. by solving differential equation for I(x): then integrate both sides.

Nonhomogeneous Linear Differential Equations

More U-Substitution February 17, Substitution Rule for Indefinite Integrals If u = g(x) is a differentiable function whose range is an interval.

3.5 Derivatives of Trig Functions. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine.

Math 3120 Differential Equations with Boundary Value Problems

Non-Homogeneous Equations

7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be.

Class Work 1.Sketch a right triangle that has acute angle , and find the five other trig ratios of . 2.Evaluate the expression without using a calculator.

Copyright © Cengage Learning. All rights reserved. 17 Second-Order Differential Equations.

Section 4.4 Undetermined Coefficients— Superposition Approach.

8.3-4 – Logarithmic Functions. Logarithm Functions.

Katie Bisciotti Alyssa Mayer Andrew Stacy

Chapter 3 Exponential and Logarithmic Functions. Chapter 3.1 Exponential Functions.

Continuity Theorems. If f and g are continuous at a and c is a constant, then the following are also continuous f + g f – g cf fg f/g if g≠0.

6.3 Dividing Polynomials 1. When dividing by a monomial: Divide each term by the denominator separately 2.

SAT Prep. Basic Differentiation Rules and Rates of Change Find the derivative of a function using the Constant Rule Find the derivative of a function.

Sec 3.1: DERIVATIVES of Polynomial and Exponential Example: Constant function.

Second-Order Differential

Sec. 3.3: Rules of Differentiation. The following rules allow you to find derivatives without the direct use of the limit definition. The Constant Rule.

Section 4.5 Undetermined coefficients— Annhilator Approach.

13.5 – The Cosine Function. I. Writing and Graphing Cosine Functions The cosine function, y = cos θ, matches the x- coordinates of the terminal side as.

Section 4.7 Variation of Parameters. METHOD OF VARIATION OF PARAMETERS For a second-order linear equation in standard form y″ + Py′ + Qy = g(x). 1.Find.

Solving Equations with Trig Functions. Labeling a right triangle A.

Chapter 5: Systems of Linear Equations Section 5.1: Solving Systems of Linear Equations by Elimination.

An IVP would look like Second Order Linear DE’s. Thm. Existence of a Unique Solution Let a 0, a 1, a 2, and g(x) be continuous on an interval containing.

AP Calculus AB Review Part One. Limits  You should be able to  Understand limits, including asymptotic and unbounded behavior  Calculate limits using.

AP Calculus 3.2 Basic Differentiation Rules Objective: Know and apply the basic rules of differentiation Constant Rule Power Rule Sum and Difference Rule.

CFA # 4 Review.

Ch 4.3: Nonhomogeneous Equations: Method of Undetermined Coefficients

SECOND-ORDER DIFFERENTIAL EQUATIONS

Lecture 8 – Integration Basics

Differentiation Product Rule By Mr Porter.

7.1 Integration By Parts.

Week 8 Second-order ODEs Second-order linear homogeneous ODEs

MATH 1330 Section 6.1.

Chapter Integration By Parts

2.7 Nonhomogeneous ODEs Section 2.7 p1.

Logs – Solve USING EXPONENTIATION

Objective - To factor trinomials in the form .

Packet #15 Exponential and Logarithmic Equations

Copyright © Cengage Learning. All rights reserved.

Before we start the lesson

MAE 82 – Engineering Mathematics

Sum and Difference Identities

Basic Differentiation Rules and Rate of Change

Sec 3.1: DERIVATIVES of Polynomial and Exponential

Ch 5.5: Multiple-Angle and Product-to-Sum Formulas

Simple Trig Equations Dr. Shildneck.

Boyce/DiPrima 9th ed, Ch 4.3: Nonhomogeneous Equations: Method of Undetermined Coefficients Elementary Differential Equations and Boundary Value Problems,

Warm-up: HW: pg. 490(1 – 4, 7 – 16, , 45 – 48)

5-3 Tangent of Sums & Differences

Objective - To factor trinomials in the form .

Copyright © 2017, 2013, 2009 Pearson Education, Inc.

MATH 1330 Section 6.1.

4.3 – Differentiation of Exponential and Logarithmic Functions

Law of Cosines C a b A B c.

Objective - To factor trinomials in the form .

Plan of the Day One-Question Quiz Score Chapter 1 Test

Section 6.5 Law of Cosines Objectives:

Week 8 Second-order ODEs Second-order linear homogeneous ODEs

8-4 Trigonometry Vocab Trigonometry: The study of triangle measurement

5-7A Classifying Zero’s Algebra II.

1.2 Essential Functions.

Presentation transcript:

Undetermined Coefficients –– Superposition Consider the problem of finding yp for the non–homogeneous DE where a0, a1, …, an are constant, and where g(x) is either polynomial, exponential, sine, cosine, or the sum or product of these.

Derivatives of functions in this group will remain in the group. This means we can assume that yp has the same form as g(x). No logarithms, no negative or fractional powers, and no other trig functions

Ex. Find yp for y – y + y = 2sin 3x

Ex. Solve y + 4y – 2y = 2x2 – 3x + 6

Ex. Solve y – 5y + 4y = 8ex

Case I: No part of the assumed yp is part of yc Look at p. 143 for several typical forms we can use Ex. Find the form of yp g(x) = 5x3e–x – 7ex g(x) = xcos x g(x) = 3x2 – 5sin 2x + 7xe6x

Case II: A part of the assumed yp is part of yc Ex. What is the form of yp for y – 6y + 9y = 6x2 + 2 – 12e3x  Multiply by enough x’s so that yp is no longer part of yc

Ex. Solve y + y = 4x + 10sin x, y(π) = 0, y(π) = 2.

Ex. Find the form of yp for y(4) + y = 1 – x2e–x