Preliminary: Linear Equations
Homogeneous Linear Equations with Constant Coefficients
Summary
Chapter 4: Higher-Order Differential Equations –
Part 1
王奕翔
Department of Electrical Engineering
National Taiwan University
October 8, 2013
王奕翔 DE Lecture 5
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Higher ordered differential equation, Lecture notes of Statistics
Linear higher-order differential equations, initial-value and boundary-value problems, homogeneous and nonhomogeneous equations, and the superposition principle. It also covers linear dependence and independence of functions. examples and theorems to explain the concepts. a lecture summary from the Department of Electrical Engineering at National Taiwan University.
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Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary
Chapter 4: Higher-Order Differential Equations –
Part 1
Department of Electrical Engineering National Taiwan University [email protected]
October 8, 2013
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations 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 equations, where n 2.
a n (x)
d n y dx n^
- a n 1 (x)
d n ^1 y dx n ^1
- + a 1 (x)
dy dx
- a 0 (x)y = g(x) (1)
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary 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
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Initial-Value Problem (IVP)
An n-th order initial-value problem associate with (1) takes the form:
Solve:
a n (x) d n y dx n^
a n 1 (x) d n ^1 y dx n ^1
+ a 1 (x) dy dx
a 0 (x)y = g(x) (1)
subject to:
y(x 0 ) = y 0 , y′(x 0 ) = y 1 ,... , y( n 1)(x 0 ) = y n 1 (2)
Here (2) is a set of initial conditions.
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Boundary-Value Problem (BVP)
Example (Second-Order ODE) Consider the following second-order ODE
a 2 (x)
d^2 y dx^2
- a 1 (x)
dy dx
- a 0 (x)y = g(x) (3)
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.
BVP: solve (3) s.t.
α 1 y(a) + β 1 y′(a) = γ 1 α 2 y(b) + β 2 y′(b) = γ 2
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Existence and Uniqueness of the Solution to an IVP
Solve
a n (x)
d n y dx n^
- a n 1 (x)
d n ^1 y dx n ^1
- + a 1 (x)
dy dx
- a 0 (x)y = g(x) (1)
subject to
y(x 0 ) = y 0 , y′(x 0 ) = y 1 ,... , y( n 1)(x 0 ) = y n 1 (2)
Theorem 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 2 I , then the above IVP has a unique solution in I_._
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Existence and Uniqueness of the Solution to an BVP
Note : Unlike an IVP, even the n-th order ODE (1) satisfies the conditions in the previous theorem, a BVP corresponding to (1) may have many, one, or no solutions.
Example
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 to the following boundary conditions respectively y(0) = 0, y(2π) = 0 Plug it in =) c 1 = 0, c 1 = 0 =) c 2 is arbitrary =) infinitely many solutions! y(0) = 0, y(π/2) = 0 Plug it in =) c 1 = 0, c 2 = 0 =) c 1 = c 2 = 0 =) a unique solution! y(0) = 0, y(2π) = 1 Plug it in =) c 1 = 0, c 1 = 1 =) contradiction =) no solutions!
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary 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
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Differential Operators
We introduce a differential operator D, which simply represent the operation of taking an ordinary differentiation:
Differential Operator For a function y = f(x), the differential operator D transforms the function f(x) to its first-order derivative: Dy :=
dy dx
Higher-order derivatives can be represented compactly with D as well:
d^2 y dx^2
= D(Dy) =: D^2 y,
d n y dx n^
=: D n y
a n (x)
d n y dx n^
- a n 1 (x)
d n ^1 y dx n ^1
- + a 1 (x)
dy dx
- a 0 (x)y =:
{ (^) n ∑
i =
a i (x)D i
y
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Differential Operators and Linear Differential Equations
Note : Polynomials of differential operators are differential operators.
Let L :=
∑ n i =0 a i (x)D
i (^) be an n-th order differential operator.
Then we can compactly represent the linear differential equation (1) and the homogeneous linear DE (4) as
L(y) = g(x), L(y) = 0
respectively.
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Linear Dependence and Independence of Functions
In Linear Algebra, we learned that one can view the collection of all functions defined on a common interval as a vector space , where linear dependence and independence can be defined respectively.
Definition (Linear Dependence and Independence) A set of functions ff 1 (x), f 2 (x),... , f n (x)g are linearly dependent on an interval I if 9 c 1 , c 2 ,... , c n not all zero such that
c 1 f 1 (x) + c 2 f 2 (x) + + c n f n (x) = 0, 8 x 2 I,
that is, the linear combination is a zero function. If the set of functions is not linearly dependent, it is linearly independent.
Example : f 1 (x) = sin^2 x, f 2 (x) = cos^2 x, I = ( π, π): Linearly dependent f 1 (x) = 1, f 2 (x) = x, f 3 (x) = x^3 , I = R: Linearly independent.
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Linear Independence of Solutions to (4)
Consider the homogeneous linear n-th order DE
a n (x)
d n y dx n^
- a n 1 (x)
d n ^1 y dx n ^1
- + a 1 (x)
dy dx
- a 0 (x)y = 0, (4)
Given n solutions ff 1 (x), f 2 (x),... , f n (x)g, we would like to test if they are independent or not. Of course we can always go back to the definition but it is clumsy... Recall : In Linear Algebra, to test if n vectors f v 1 , v 2 ,... , v n g are linearly independent, we can compute the determinant of the matrix
V :=
[
v 1 v 2 v n
]
If det V = 0, they are linearly dependent; if det V ̸= 0, they are linearly independent.
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
Fundamental Set of Solutions
We are interested in describing the solution space , that is, the subspace spanned by the solutions to the homogeneous linear n-th order DE
a n (x) d n y dx n^
a n 1 (x) d n ^1 y dx n ^1
+ a 1 (x) dy dx
a 0 (x)y = 0. (4)
How? Recall : In Linear Algebra, we describe a subspace by its basis : any vector in the subspace can be represented by a linear combination of the elements in the basis, and these elements are linearly independent. Similar things can be done here.
Definition (Fundamental Set of Solutions) Any set ff 1 (x), f 2 (x),... , f n (x)g of n linearly independent solutions to the homogeneous linear n-th order DE (4) on an interval I is called a fundamental set of solutions.
Homogeneous Linear Equations with Constant CoefficientsPreliminary: Linear Equations Summary Nonhomogeneous Equations
General Solutions to Homogeneous Linear DE
General solution to an n-th order ODE: An n-parameter family of solutions that can contains all solutions.
Theorem Let ff 1 (x), f 2 (x),... , f n (x)g be a fundamental set of solutions to the homogeneous linear n -th order DE (4) on an interval I_. Then the_ general solution to (4) is
y = c 1 f 1 (x) + c 2 f 2 (x) + + c n f n (x),
where fc i j i = 1, 2 ,... , ng are arbitrary constants.