Chapter 4: Higher-Order Differential Equations Part 1 Department - PowerPoint PPT Presentation

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Summary Chapter 4: Higher-Order Differential Equations – Part 1 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 8, 2013 DE Lecture 5 王奕翔 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Summary Higher-Order Differential Equations Most of this chapter deals with linear higher-order DE (except 4.10) In our lecture, we skip 4.10 and focus on n -th order linear differential (1) DE Lecture 5 equations, where n ≥ 2 . dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Summary Methods of Solving Linear Differential Equations We shall gradually fill up this slide as the lecture proceeds. DE Lecture 5 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Summary Initial-Value and Boundary-Value Problems Homogeneous Equations Nonhomogeneous Equations 1 Preliminary: Linear Equations Initial-Value and Boundary-Value Problems Homogeneous Equations Nonhomogeneous Equations 2 Homogeneous Linear Equations with Constant Coefficients Second Order Equations n -th Order Equations 3 Summary DE Lecture 5 王奕翔

Preliminary: Linear Equations Solve: Here (2) is a set of initial conditions . (2) subject to: (1) Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 An n -th order initial-value problem associate with (1) takes the form: Initial-Value Problem (IVP) Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 1 , . . . , y ( n − 1) ( x 0 ) = y n − 1 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Remark (Order of the derivatives in the conditions specify an unique solution. “Usually” a n -th order ODE requires n initial/boundary conditions to Remark (Number of Initial/Boundary Conditions) Boundary Conditions: conditions can be at different x . DE Lecture 5 Remark (Initial vs. Boundary Conditions) Recall: in Chapter 1, we made 3 remarks on initial/boundary conditions Boundary-Value Problem (BVP) Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Initial Conditions: all conditions are at the same x = x 0 . Initial/boundary conditions can be the value or the function of 0 -th to ( n − 1) -th order derivatives, where n is the order of the ODE. 王奕翔

Preliminary: Linear Equations Example (Second-Order ODE) BVP: solve (3) s.t. (3) Homogeneous Linear Equations with Constant Coefficients Consider the following second-order ODE DE Lecture 5 Boundary-Value Problem (BVP) Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary a 2 ( x ) d 2 y dx 2 + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) IVP: solve (3) s.t. y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 1 . BVP: solve (3) s.t. y ( a ) = y 0 , y ( b ) = y 1 . BVP: solve (3) s.t. y ′ ( a ) = y 0 , y ( b ) = y 1 . { α 1 y ( a ) + β 1 y ′ ( a ) = γ 1 α 2 y ( b ) + β 2 y ′ ( b ) = γ 2 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients the above IVP has a unique solution in I . Theorem (2) subject to (1) DE Lecture 5 Solve Existence and Uniqueness of the Solution to an IVP Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 1 , . . . , y ( n − 1) ( x 0 ) = y n − 1 If a n ( x ) , a n − 1 ( x ) , . . . , a 0 ( x ) and g ( x ) are all continuous on an interval I , a n ( x ) ̸ = 0 is not a zero function on I , and the initial point x 0 ∈ I , then 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Throughout this lecture, we assume that on some common interval I , (2) subject to (1) DE Lecture 5 Solve Existence and Uniqueness of the Solution to an IVP Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) y ( x 0 ) = y 0 , y ′ ( x 0 ) = y 1 , . . . , y ( n − 1) ( x 0 ) = y n − 1 a n ( x ) , a n − 1 ( x ) , . . . , a 0 ( x ) and g ( x ) are all continuous a n ( x ) is not a zero function, that is, ∃ x ∈ I such that a n ( x ) ̸ = 0 . 王奕翔

Preliminary: Linear Equations to the following boundary conditions respectively = Plug it in = = Plug it in = = Homogeneous Linear Equations with Constant Coefficients Plug it in = DE Lecture 5 Example Note : Unlike an IVP, even the n -th order ODE (1) satisfies the Summary Initial-Value and Boundary-Value Problems Homogeneous Equations Existence and Uniqueness of the Solution to an BVP Nonhomogeneous Equations conditions in the previous theorem, a BVP corresponding to (1) may have many, one, or no solutions. Consider the 2nd-order ODE d 2 y dx 2 + y = 0 , whose general solution takes the form y = c 1 cos x + c 2 sin x . Find the solution(s) to an BVP subject y (0) = 0 , y (2 π ) = 0 ⇒ c 1 = 0 , c 1 = 0 ⇒ c 2 is arbitrary = ⇒ infinitely many solutions! y (0) = 0 , y ( π /2) = 0 ⇒ c 1 = 0 , c 2 = 0 ⇒ c 1 = c 2 = 0 = ⇒ a unique solution! y (0) = 0 , y (2 π ) = 1 ⇒ c 1 = 0 , c 1 = 1 ⇒ contradiction = ⇒ no solutions! 王奕翔

Preliminary: Linear Equations Homogeneous Linear Equations with Constant Coefficients Summary Initial-Value and Boundary-Value Problems Homogeneous Equations Nonhomogeneous Equations 1 Preliminary: Linear Equations Initial-Value and Boundary-Value Problems Homogeneous Equations Nonhomogeneous Equations 2 Homogeneous Linear Equations with Constant Coefficients Second Order Equations n -th Order Equations 3 Summary DE Lecture 5 王奕翔

Preliminary: Linear Equations (1) equation, we must first solve its associated homogeneous equation (4). Later in the lecture we will see, when solving a nonhomogeneous associated homogeneous equation (4) is the one with the same (4) Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 Nonhomogeneous Equations Linear n -th order ODE takes the form: Homogeneous Equation Summary Initial-Value and Boundary-Value Problems Homogeneous Equations dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) Homogeneous Equation : g ( x ) in (1) is a zero function: dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = 0 Nonhomogeneous Equation : g ( x ) in (1) is not a zero function. Its coefficients except that g ( x ) is a zero function 王奕翔

Preliminary: Linear Equations Differential Operator y d n y Homogeneous Linear Equations with Constant Coefficients Higher-order derivatives can be represented compactly with D as well: dx . DE Lecture 5 operation of taking an ordinary differentiation: We introduce a differential operator D , which simply represent the Differential Operators Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary For a function y = f ( x ) , the differential operator D transforms the function f ( x ) to its first-order derivative: Dy := dy d 2 y dx 2 = D ( Dy ) =: D 2 y , dx n =: D n y { n } dx n + a n − 1 ( x ) d n − 1 y ∑ a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y =: a i ( x ) D i i =0 王奕翔

Preliminary: Linear Equations Note : Polynomials of differential operators are differential operators. respectively. the homogeneous linear DE (4) as Then we can compactly represent the linear differential equation (1) and Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 Differential Operators and Linear Differential Equations Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Let L := ∑ n i =0 a i ( x ) D i be an n -th order differential operator. L ( y ) = g ( x ) , L ( y ) = 0 王奕翔

Preliminary: Linear Equations Linearity and Superposition Principle (4) Homogeneous Linear Equations with Constant Coefficients the following superposition principle. Theorem (Superposition Principle: Homogeneous Equations) Nonhomogeneous Equations Homogeneous Equations Summary Initial-Value and Boundary-Value Problems DE Lecture 5 L := ∑ n i =0 a i ( x ) D i is a linear operator : for two functions f 1 ( x ) , f 2 ( x ) , L ( λ 1 f 1 + λ 2 f 2 ) = λ 1 L ( f 1 ) + λ 2 L ( f 2 ) . For any homogeneous linear equation (4), that is, L ( y ) = 0 , we obtain Let f 1 , f 2 , . . . , f k be solutions to the homogeneous n -th order linear equation L ( y ) = 0 on an interval I , that is, dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = 0 , then the linear combination f = ∑ k i =1 λ i f i is also a solution to (4) . 王奕翔