SECOND-ORDER DIFFERENTIAL EQUATIONS

Slides:

Advertisements

Similar presentations

First Order Linear Differential Equations

Advertisements

Differential Equations

STROUD Worked examples and exercises are in the text PROGRAMME F6 POLYNOMIAL EQUATIONS.

Differential Equations MTH 242 Lecture # 11 Dr. Manshoor Ahmed.

A second order ordinary differential equation has the general form

Ch 3.3: Complex Roots of Characteristic Equation Recall our discussion of the equation where a, b and c are constants. Assuming an exponential soln leads.

Linear Equations Please Read Entire Section 2.1, pages in the text.

Ch 3.4: Repeated Roots; Reduction of Order

Ch 3.5: Repeated Roots; Reduction of Order

Math for CS Second Order Linear Differential Equations

2 nd Order Differential Equations Type 1 click for link Type 2click for link Exceptions to the Particular Integral Rule 1 st order equations which can.

1 Chapter 9 Differential Equations: Classical Methods A differential equation (DE) may be defined as an equation involving one or more derivatives of an.

PARTIAL DIFFERENTIAL EQUATIONS

Boyce/DiPrima 9 th ed, Ch 3.1: 2 nd Order Linear Homogeneous Equations-Constant Coefficients Elementary Differential Equations and Boundary Value Problems,

STROUD Worked examples and exercises are in the text PROGRAMME F7 PARTIAL FRACTIONS.

ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

U4L3 Solving Quadratic Equations by Completing the Square.

Nonhomogeneous Linear Differential Equations

Solve.. Question of the Day CCGPS Geometry Day 62 ( ) UNIT QUESTION: How are real life scenarios represented by quadratic functions? Today’s.

Fin500J Topic 6Fall 2010 Olin Business School 1 Fin500J: Mathematical Foundations in Finance Topic 6: Ordinary Differential Equations Philip H. Dybvig.

Differential Equations MTH 242 Lecture # 13 Dr. Manshoor Ahmed.

SOLVING QUADRATIC EQUATIONS Unit 7. SQUARE ROOT PROPERTY IF THE QUADRATIC EQUATION DOES NOT HAVE A “X” TERM (THE B VALUE IS 0), THEN YOU SOLVE THE EQUATIONS.

Mathematics. Session Differential Equations - 2 Session Objectives  Method of Solution: Separation of Variables  Differential Equation of first Order.

UNIVERSAL COLLEGE OF ENGINEERING AND TECHNOLOGY. FIRST ORDER LINEAR DIFFERENTIAL EQUATION PRESENTED BY ANVI VIRANI ENROLL NO:

8-1 Completing the Square

12/19/ Non- homogeneous Differential Equation Chapter 4.

Second-Order Differential

CSE 2813 Discrete Structures Solving Recurrence Relations Section 6.2.

2.1 – Linear and Quadratic Equations Linear Equations.

1 (x  1) 2 = 8 2x + 3y = 11 A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers.

Math 3120 Differential Equations with Boundary Value Problems

9.1c SKM & PP 1 Quadratic Equations. 9.1c SKM & PP 2 Square Roots: Know Your Squares! Consider this equation: What number(s) could be squared to give.

Non-Homogeneous Second Order Differential Equation.

Differential Equations Linear Equations with Variable Coefficients.

PreCalculus Section 1.6 Solve quadratic equations by: a. Factoring b. Completing the square c. Quadratic formula d. Programmed calculator Any equation.

Worked examples and exercises are in the text STROUD PROGRAMME 24 FIRST-ORDER DIFFERENTIAL EQUATIONS.

STROUD Worked examples and exercises are in the text Programme 25: First-order differential equations FIRST-ORDER DIFFERENTIAL EQUATIONS PROGRAMME 25.

Differential Equations MTH 242 Lecture # 28 Dr. Manshoor Ahmed.

5.9 Complex Numbers Alg 2. Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.

STROUD Worked examples and exercises are in the text Programme F8: Partial fractions PROGRAMME F8 PARTIAL FRACTIONS.

Lecture 5 Inhomogeneous ODEs Intro to Fourier Series You’ll solve an inhomogeneous ODE extra example We’ll look at using complex numbers to solve IODEs.

Formation of Partial Differential equations Partial Differential Equation can be formed either by elimination of arbitrary constants or by the elimination.

First Order Linear Differential Equations Any equation containing a derivative is called a differential equation. A function which satisfies the equation.

STROUD Worked examples and exercises are in the text 1 STROUD Worked examples and exercises are in the text Programme F5: Linear equations PROGRAMME F5.

8: Simultaneous Equations and Intersections © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

Section 2.5 – Quadratic Equations

PreCalculus Section 1. 6 Solve quadratic equations by: a. Factoring b

Welcome! Grab a set of interactive notes

Linear homogeneous ODEn with constant coefficients

SECONDD ORDER LINEAR DIFFERENTIAL EQUATIONS

Linear Equations Constant Coefficients

Engineering Analysis I

DIFFERENTIAL EQUATIONS

The exact values can be found by solving the equations simultaneously

Equations Quadratic in form factorable equations

Solving Equations by Factoring

We will be looking for a solution to the system of linear differential equations with constant coefficients.

A second order ordinary differential equation has the general form

MAE 82 – Engineering Mathematics

A quadratic equation is written in the Standard Form,

“Teach A Level Maths” Vol. 1: AS Core Modules

Solving Quadratic Equations by Factoring

Systems of Linear and Quadratic Equations

Standard Form Quadratic Equation

Cauchy–Euler Equation

Systems of Equations Solve by Graphing.

4.5: Completing the square

Equations Quadratic in form factorable equations

PARTIAL DIFFERENTIAL EQUATIONS

Differential Equations

Presentation transcript:

SECOND-ORDER DIFFERENTIAL EQUATIONS PROGRAMME 25 SECOND-ORDER DIFFERENTIAL EQUATIONS

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations Introduction For any three numbers a, b and c, the two numbers: are solutions to the quadratic equation: with the properties:

Programme 25: Second-order differential equations Introduction The differential equation: can be re-written to read: that is:

Programme 25: Second-order differential equations Introduction The differential equation can again be re-written as: where:

Programme 25: Second-order differential equations Introduction The differential equation: has solution: This means that: That is:

Programme 25: Second-order differential equations Introduction The differential equation: has solution: where:

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations Homogeneous equations The differential equation: Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation: These are:

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations The auxiliary equation Real and different roots Real and equal roots Complex roots

Programme 25: Second-order differential equations The auxiliary equation Real and different roots If the auxiliary equation: with solution: where: then the solution to:

Programme 25: Second-order differential equations The auxiliary equation Real and equal roots If the auxiliary equation: with solution: where: then the solution to:

Programme 25: Second-order differential equations The auxiliary equation Complex roots If the auxiliary equation: with solution: where: Then the solutions to the auxiliary equation are complex conjugates. That is:

Programme 25: Second-order differential equations The auxiliary equation Complex roots Complex roots to the auxiliary equation: means that the solution of the differential equation: is of the form:

Programme 25: Second-order differential equations The auxiliary equation Complex roots Since: then: The solution to the differential equation whose auxiliary equation has complex roots can be written as::

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations Summary Differential equations of the form: Auxiliary equation: Roots real and different: Solution Roots real and the same: Solution Roots complex (  j): Solution

Programme 25: Second-order differential equations Introduction Homogeneous equations The auxiliary equation Summary Inhomogeneous equations

Programme 25: Second-order differential equations Inhomogeneous equations The second-order, constant coefficient, linear, inhomogeneous differential equation is an equation of the type: The solution is in two parts y1 + y2: part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation part 2, y2 is called the particular integral.

Programme 25: Second-order differential equations Inhomogeneous equations Complementary function Example, to solve: Complementary function Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3 Complementary function y1 = Ae2x + Be3x where:

Programme 25: Second-order differential equations Inhomogeneous equations Particular integral (b) Particular integral Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in: gives: yielding: so that:

complementary function + particular integral Programme 25: Second-order differential equations Inhomogeneous equations Complete solution (c) The complete solution to: consists of: complementary function + particular integral That is:

Programme 25: Second-order differential equations Inhomogeneous equations Particular integrals The general form assumed for the particular integral depends upon the form of the right-hand side of the inhomogeneous equation. The following table can be used as a guide:

Programme 25: Second-order differential equations Learning outcomes Use the auxiliary equation to solve certain second-order homogeneous equations Use the complementary function and the particular integral to solve certain second-order inhomogeneous equations