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# Week 13 Notes Math 340

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This 4 page Class Notes was uploaded by Susan Ossareh on Sunday April 24, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 7 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 04/24/16

th Math 340 Lecture – Introduction to Ordinary Differential Equations – April 18 , 2016 What We Covered: 1. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.9: Inhomogeneous Linear Systems i. Definition: You’re given the linear equation ???? = ???? ???? ???? + ????(????) where f(t) is the inhomogeneous term because it’s not dependent on y ii. Theorem: Suppose that y is p particular solution to the inhomogeneous equation and that ???? 1???? 2...,???????????? ???? f????rm a fundamental set of solutions to the associated ′ homogeneous equation ???? = ???? ???? ????. Then the general solution to the inhomogeneous equation is given by ???? = ???? + ???? ???? + ???? ???? +...+ ???? ???? , where C , ???? 1 1 2 2 ???? ???? 1 C2, and C nre arbitrary constants 1. This basically means you have the equation ???? = ???? + ???? s???? you????need to set up y like you usually would, where you find the eigenvectors and create n the general equation. Then you’re goal in section 9.9 is to find y p iii. Fundamental Matrices 1. We start with a fundamental set of solutions to the associated homogeneous system. Let Y(t) be the nxn matrix whose ith column is ???? ???? ,???????????? 1 ≤ ???? ≤ ????. Thus ???? = [???? ,???? ,...,???? ]. A matrix like Y, whose ???? 1 2 ???? columns form a fundamental set of solutions for the system is called a fundamental matrix. 2. Since ????′????= ???????? f????r each i, we have ???? = [???? ,???? ,...,???? ] = ′ 1 2 ???? [????????1,???????? 2...,???????? ????] = ???????? iv. Proposition: A matrix-valued function Y(t) is a fundamental matrix for the system y’=Ay if and only if Y’=AY and Y =Y(0 ) is0invertible for some t 0 v. Variation of Parameters for Systems 1. To find a solution to the inhomogeneous equations, we want it in the ???? form ???? ???? = ???? ???? ????(????), where ???? ???? = (???? ???? ,...1???? (????)) is????a column vector of functions to be determined, and ???? = [???? ,???? 1..2,???? ] i???? a fundamental ????1 [ ] ⋮ matrix. Using matrix multiplication, ???? = ???? ,???? ,.1.,2 ???? ∗ ( ) = ???????? ???? ???? +...+???? ???? 1 1 ???? ???? a. We are then lead to know a few things based off of this definition −1 i. ???? = ∫ ???? (???? ???? ???? ???????? ii. ???? = ????(????) ∫ ???? −1(???? ???? ???? ???????? ???? iii. And if we need to find y with an initial condition y(t )00 ???? −1 then we can solve: ???? ???? ????(????) ∫???? ???? (???? ???? ???? ???????? 0 iv. And if we are given an initial condition y(t )0y t0en we −1 get ???? = ???? (????0)????0 vi. Theorem: Suppose that A is a real nxn matrix and that Y(t) is a fundamental matrix for the system. Let f(t) be a vector-valued function. Then the solution to the initial value problem ???? = ???????? + ???? ????????????ℎ ???? ???? ( ) = ???? ′ 0 0 vii. To find the general solution to ???? = ???????? + ???? 1. Find Y(t) a. Solve y’=Ay b. Find the fundamental set of solutions c. Find the eigenvalues and eigenvectors −1 2. Find ???? (????) 3. Find ???? = ????(????) ∫ ???? −1(???? ???? ???? ???????? ???? Suggested Homework: Section 9.9: 2, 4, 6, 8, 20, 22, 26, 28 th Math 340 Lab – Introduction to Ordinary Differentia Equations – April 24 , 2016 What We Covered: 1. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.9 continued: Inhomogeneous Linear Systems i. To solve y’=Ay+f 1. Find Y(t) which is basically y n 2. Find Y (t) 3. Find y p ii. Initial condition theorem −1 1. ????????= ????(????) ∫???? (???? ???? ???? ???????? −1 ???? −1 2. ???? ???? = ????(????)(???? (????0???? 0 ∫????0???? (???? ???? ???? ???????? iii. So you essentially have two ways to solve an inhomogeneous linear system 1. Method #1: find the constants C 2. Method #2: use the IC theorem 2. Course Content – Chapter 4: Second Order Equations a. Section 4.5: Inhomogeneous Equations; the Method of Undetermined Coefficients i. Method of Undetermined Coefficients 1. So we want to look at the inhomogeneous equation, it can only work if the coefficients are constant 2. The method of undetermined coefficients is based on the fact that there are some situations where the form of the forcing term allows us to almost guess the form of a particular solution 3. If the forcing term f has a for that is replicated under differentiation, then look for a solution with the same general form as the forcing term ii. Exponential Forcing Terms ???????? ′ ???????? 1. Where the forcing term is ???? ???? = ???? ,????ℎ???????? ???? ???? = ???????? ) 2. Example: Find a particular solution to the equation ???? − ???? − 2???? = 2???? −2???? Our forcing term is ???? ???? = 2???? −2????, when we take the derivative, a is an undetermined coefficient so… ???? ???? = −2???????? −2???? ???????????? ???? ′(???? = 4???????? −2???? ???????? ????ℎ???????? ???????? ???????????? ???????????????????????? ????ℎ???????????? ???????????????? ????ℎ???? ???????????????????? ???????????????????????????????? … ???? − ???? − 2???? = 4???????? −2????− −2???????? −2????)− 2 ???????? −2????) = 4???????? −2???? −2???? −2???? 4???????? = 2???? ???????????????????? ????ℎ???? ???????????????????????????????????????????????? … −2???? ???? ???? = ???? 2 So then the general solution is… 1 −2???? −???? 2???? ???? ???? = ???? + ????1???? + ???? 2 2 Suggested Homework: Section 9.9: 2, 4, 6, 8, 20, 22, 26, 28 Section 4.5: 7, 12, 18, 20, 22, 26, 41, 43 Math 340 Lecture – Introduction to Ordinary Differential Equations – April 20 , 2016 th What We Covered: 1. Quiz on 9.9 2. Course Content – Chapter 4 : Second Order Equations a. Section 4.5 continued: Inhomogeneous Equation i. Trigonometric Forcing Terms 1. Consider a forcing term of the form ???? ???? = ???????????????? ???????? + ????????????????(????????). The derivative of f has the same general form, so we will look for solutions of the form ???? ???? = acos ???????? + ???????????????? ???????? , where a and b are as a yet determined coefficients ii. Example: Given ???? + 7???? + 10???? = −4sin(3????), fine ???? ???? = acos 3???? + ????????????????(3????) ???? ????′????= −3asin 3???? + 3????????????????(3????) ????′′ = −9acos 3???? − 9????????????????(3????) ???? ????′???? + 7????′????+ 10???? =????−4sin(3????) −9acos 3???? − 9???????????????? 3???? + 7(−3asin 3???? + 3???????????????? 3???? ) + ???? sin 3???? ( ( )) = −4????????????3???? (−9a+21b+10a)cos(3t)+(-9b-21a+10b)sin(3t)=-4sin(3t) (???? + 21???? cos 3???? + ???? − 21???? sin 3???? = −4sin 3???? + 0cos(3????) ???????????????????????????? ???????????????????????????????????????????????? ???????????? ???????????????????? ???????????? ???? ???????????? ????: cos 3???? : ???? + 21???? = 0 sin 3???? :???? − 21???? = −4 ???? = −21???? ???? − 21 −21???? = −4 1 21 ???? = − ???? = 110 110 21 1 ????????= ????????????3???? − ????????????3???? 110 110 Suggested Homework: Section 4.5: 7, 12, 18, 20, 22, 26, 41, 43 Math 340 Lecture – Introduction to Ordinary Differential Equations – April 22 , 2016 nd What We Covered: 1. Course Content – Chapter 4: Second Order Equations a. Section 4.5 continued: Exceptional cases i. The undetermined coefficients looks like a straightforward method where the forcing term is a solution to the associated homogeneous equation. So what do we −???? do? We look for a solution of the form ???? ???? = ???????????? . So we can , multiply the usual general form by t. this is the way to find a solution whenever the usual form doesn’t work ii. Theorem: Given ′′ ′ 1. ???? + ???????? + ???????? = ???? 2. ???? + ???????? + ???????? = ???? 3. Let yfbe a solution for 1 and y ge a solution for 2 4. Then you can find the solution for ???? + ???????? + ???????? = ???????? + ???????? in the form ???? = ???????????? + ???????????? iii. More complicated forcing terms FORCING TERM TRIAL SOLUTION ???????? ???????? ???? ???????? ???? ???????? ???????? ???????? ???? ????(????) ????(????) ???????????? ???????? ???????? sin(????????) ???????????????? ???????? + ????sin(????????) ???? ???? ???????????? ???????? ???????? ???? ???? sin(????????) ???? ???? cos ???????? + ???? ???? sin(????????) ???? ???????????? ???????? ???????? ???? sin(????????) ???? [???????????????? ???????? + ???? sin ???????? ] ) ???????? ???????? ???????? ???? ???? ???? ???????????? ???????? ???????? ???? ???? ???? sin(????????) ???? [???? ???? cos ???????? + ???? ???? sin ???????? ]( ) 2. Course Content – Chapter 5: The Laplace Transform a. Section 5.1: The Definition of the Laplace Transform i. Definition: Suppose f(t) is a function of t defined for 0 < ???? < ∞, The Laplace transform of f is the function: ∞ ℒ ???? ???? = ???? ???? =( ) ∫ ???? ???? ????−???????? ???????? for ???? > 0 0 Suggested Homework: Section 4.5: 7, 12, 18, 20, 22, 26, 41, 43

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