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A Second Course in Elementary Ordinary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved
1
Contents 26 Calc Calcul ulus us of Matr Matrix ix-V -Val alue ued d Fun unct ctio ions ns of a Real Real Varia ariabl ble e
4
27 nth Order Linear Differential Equations:Existence and Uniqueness 18 28 The General Solution of nth Order Linear Homogeneous Equations 25 29 Fundamental Sets and Linear Indep endence
34
30 Higher Order Homogeneous Linear Equations with Constant Co efficients 40 31 Non Homogeneous Homogeneous nth Order Linear Differenti Differential al Equations Equations 47 32 Existence and Uniqueness of Solution to Initial Value First Order Linear Systems 56 33 Homogeneous First Order Linear Systems
63
34 First First Order Linear Linear Systems: Systems: Fundamen undamental tal Sets and Linear Indep endence 74 35 Homogeneous Systems with Constant Coeffi oefficients
83
36 Homogeneous Systems with Constant Coefficients: Complex Eigenvalues 93 37 Homogeneous Systems with Constant Coefficients: Repeated Eigenvalues 97 38 Nonhomogeneous First Order Linear Systems
108
39 Solving First Order Linear Systems with Diagonalizable Constant Coefficients Matrix 118 40 Solving First Order Linear Systems Using Exponential Matrix 126 2
Contents 26 Calc Calcul ulus us of Matr Matrix ix-V -Val alue ued d Fun unct ctio ions ns of a Real Real Varia ariabl ble e
4
27 nth Order Linear Differential Equations:Existence and Uniqueness 18 28 The General Solution of nth Order Linear Homogeneous Equations 25 29 Fundamental Sets and Linear Indep endence
34
30 Higher Order Homogeneous Linear Equations with Constant Co efficients 40 31 Non Homogeneous Homogeneous nth Order Linear Differenti Differential al Equations Equations 47 32 Existence and Uniqueness of Solution to Initial Value First Order Linear Systems 56 33 Homogeneous First Order Linear Systems
63
34 First First Order Linear Linear Systems: Systems: Fundamen undamental tal Sets and Linear Indep endence 74 35 Homogeneous Systems with Constant Coeffi oefficients
83
36 Homogeneous Systems with Constant Coefficients: Complex Eigenvalues 93 37 Homogeneous Systems with Constant Coefficients: Repeated Eigenvalues 97 38 Nonhomogeneous First Order Linear Systems
108
39 Solving First Order Linear Systems with Diagonalizable Constant Coefficients Matrix 118 40 Solving First Order Linear Systems Using Exponential Matrix 126 2
41 The The Lapl Laplac ace e Trans ransfo form rm:: Basic Basic Defin Definit itio ions ns an and d Resu Result ltss
131 131
42 Further Studies of Laplace Transform
142
43 The Laplace Transform and the Method of Partial Fractions155 44 Laplace Transforms of Periodic Functions
162
45 Solving Systems of Differential Equations Using Laplace Transform 171 46 Convolution Integrals
178
47 The Dirac Delta Function and Impulse Respon ponse
186
48 Numerical Solutions to ODEs: Euler’s Method and its Variants 194
3
26
Calcul Calculus us of Matr Matrixix-V Valued alued Func Functio tions ns of a Real Variable
In establishing the existence result for second and higher order linear differential equations one transforms the equation into a linear system and tries to solve solve such such a syste system. m. This This procedu procedure re require requiress the use of concepts concepts such such as the derivative of a matrix whose entries are functions of t, t, the integral of a matrix matrix,, and the exponen exponentia tiall matrix matrix functio function. n. Thus, Thus, techni technique quess from from matrix theory play an important role in dealing with systems of differential equations. equations. The present present section section introduces introduces the necessary necessary background background in the calculus calculus of matrix matrix functions. functions.
Matrix-Valued Functions of a Real Variable A matrix A of dimension m n is a rectangular array of the form
×
A =
a11 a12 a21 a22 ... ... am1 am2
.. ... a1n .. ... a2n ... ... .. ... amn
where the aij ’s are the entries of the matrix, m is the number of rows, n is the number number of column columns. s. The zero matrix 0 is the matrix whose entries are are all all 0. The The n n identity matrix In is a square matrix whose main diagonal diagonal consists of 1 s and the off diagonal entries entries are all 0. A matrix A can be represented with the following compact notation A = (aij ). The entry a entry a ij is located in the ith row and jth column.
×
Example 26.1 Consider Consider the matrix matrix A(t) = Find a Find a 22 , a32 , and a 23 .
− −
5 10 5
0 2 2
−
− 1 0 7
Solution. The entry a22 is in the second row and second column so that a22 = Similarly, a Similarly, a 32 = 2 and a and a 23 = 0
4
−2.
An m n array whose entries are functions of a real variable defined on a common interval is called a matrix function. Thus, the matrix
is a 3 1 matrix function also known as a vector-valued function. We will denote an m n matrix function by A(t) = (aij (t)) where aij (t) is the entry in the ith row and jth coloumn.
×
×
Arithmetic of Matrix Functions All the familiar rules of matrix arithmetic hold for matrix functions as well. (i) Equality: Two m n matrices A(t) = (aij (t)) and B(t) = (bij (t)) are said to be equal if and only if a a ij (t) = b ij (t) for all 1 i m and 1 j n. That is, two matrices are equal if and only if all corresponding elements are equal. Notice that the matrices must be of the same dimension.
×
≤ ≤
≤ ≤
Example 26.2 Solve the following matrix equation for a,b, for a,b, c, and d
a b b+c = 3d + c 2a 4d
−
−
8 1 7 6
Solution. Equating Equating corresponding corresponding entries entries we get the system
a
2a
−
b b + c c + 3d 4d
−
= = = =
8 1 7 6
Adding the first two equations to obtain a + c = 9. Adding 4 times the third equation to 3 times the last equation to obtain 6a 6 a + 4c = 46 or 3a 3a + 2c = 23. 23. 5
Contents 26 Calc Calcul ulus us of Matr Matrix ix-V -Val alue ued d Fun unct ctio ions ns of a Real Real Varia ariabl ble e
4
27 nth Order Linear Differential Equations:Existence and Uniqueness 18 28 The General Solution of nth Order Linear Homogeneous Equations 25 29 Fundamental Sets and Linear Indep endence
34
30 Higher Order Homogeneous Linear Equations with Constant Co efficients 40 31 Non Homogeneous Homogeneous nth Order Linear Differenti Differential al Equations Equations 47 32 Existence and Uniqueness of Solution to Initial Value First Order Linear Systems 56 33 Homogeneous First Order Linear Systems
63
34 First First Order Linear Linear Systems: Systems: Fundamen undamental tal Sets and Linear Indep endence 74 35 Homogeneous Systems with Constant Coeffi oefficients
83
36 Homogeneous Systems with Constant Coefficients: Complex Eigenvalues 93 37 Homogeneous Systems with Constant Coefficients: Repeated Eigenvalues 97 38 Nonhomogeneous First Order Linear Systems
108
39 Solving First Order Linear Systems with Diagonalizable Constant Coefficients Matrix 118 40 Solving First Order Linear Systems Using Exponential Matrix 126 2
