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

Preliminary: Linear Equations Reduction of Order Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation Summary Chapter 4: Higher-Order Differential Equations – Part 1 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 10, 2013 DE Lecture 5 王奕翔 王奕翔

Preliminary: Linear Equations Reduction of Order Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation 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 Reduction of Order Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation 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 Reduction of Order 5 Summary 4 Cauchy-Euler Equation n -th Order Equations Second Order Equations 3 Homogeneous Linear Equations with Constant Coefficients 2 Reduction of Order Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems 1 Preliminary: Linear Equations Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 王奕翔

Preliminary: Linear Equations An n -th order initial-value problem associate with (1) takes the form: Here (2) is a set of initial conditions . (2) subject to: (1) Reduction of Order Solve: DE Lecture 5 Initial-Value Problem (IVP) Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients 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 Recall: in Chapter 1, we made 3 remarks on initial/boundary conditions 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 . Reduction of Order Remark (Initial vs. Boundary Conditions) Boundary-Value Problem (BVP) Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 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 Boundary-Value Problem (BVP) BVP: solve (3) s.t. (3) Reduction of Order Consider the following second-order ODE Example (Second-Order ODE) DE Lecture 5 Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients 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 Solve the above IVP has a unique solution in I . Theorem (2) subject to (1) Reduction of Order DE Lecture 5 Existence and Uniqueness of the Solution to an IVP Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients 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 Solve Throughout this lecture, we assume that on some common interval I , (2) subject to (1) Reduction of Order DE Lecture 5 Existence and Uniqueness of the Solution to an IVP Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients 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 have many, one, or no solutions. = Plug it in = = Plug it in = = Reduction of Order to the following boundary conditions respectively Example Plug it in = conditions in the previous theorem, a BVP corresponding to (1) may Homogeneous Equations Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation Summary Note : Unlike an IVP, even the n -th order ODE (1) satisfies the Initial-Value and Boundary-Value Problems Nonhomogeneous Equations Existence and Uniqueness of the Solution to an BVP DE Lecture 5 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 Reduction of Order 5 Summary 4 Cauchy-Euler Equation n -th Order Equations Second Order Equations 3 Homogeneous Linear Equations with Constant Coefficients 2 Reduction of Order Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems 1 Preliminary: Linear Equations Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 王奕翔

Preliminary: Linear Equations Linear n -th order ODE takes the form: 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) Reduction of Order (1) Homogeneous Equation Nonhomogeneous Equations Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation Summary Initial-Value and Boundary-Value Problems DE Lecture 5 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 operation of taking an ordinary differentiation: y d n y Reduction of Order Higher-order derivatives can be represented compactly with D as well: dx . Differential Operator DE Lecture 5 We introduce a differential operator D , which simply represent the Differential Operators Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Homogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equation 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 Differential Operators and Linear Differential Equations respectively. the homogeneous linear DE (4) as Then we can compactly represent the linear differential equation (1) and Reduction of Order Note : Polynomials of differential operators are differential operators. Nonhomogeneous Equations Homogeneous Equations Initial-Value and Boundary-Value Problems Summary Cauchy-Euler Equation Homogeneous Linear Equations with Constant Coefficients DE Lecture 5 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