2 Chapter 6 Systems of First Order Linear Equations we build on elementary theory and solution techniques for first order linear systemsScience and engineering applications that possess even a modest degree of complexity often lead to systems of differential equations of dimensionn > 2.A large portion of Chapter 6 generalizes the eigenvalue method presented in Chapter 3 to constant coefficient linear systems of dimension n > 2.
3 Chapter 6 - Systems of First Order Linear Equations 6.1 Definitions and Examples6.2 Basic Theory of First Order Linear Systems6.3 Homogeneous Linear Systems with Constant Coefficients6.4 Nondefective Matrices with Complex Eigenvalues6.5 Fundamental Matrices and the Exponential of a Matrix6.6 Nonhomogeneous Linear Systems6.7 Defective Matrices
4 6.1 Definitions and Examples Matrix-Valued Functions.The principal mathematical objects involved in the study of linear systems of differential equations are matrix-valued functions, or simply matrix functions. Theseobjects are vectors or matriceswhose elements are functions of t.We writeThe matrix P = P(t) is said to be continuous at t = t0 or on an interval I = (α, β) if each element of P is a continuous function at the given point or on the given interval. Similarly, P(t) is said to be differentiable if each of its elements is differentiable, and its derivative dP/dt is defined by Similarly, the integral of amatrix function is defined as
5 First Order Linear Systems: General Framework The general form of a first order linear system of dimension n isx'1= p11(t)x1 + p12(t)x2 + ・・・ + p1n(t)xn + g1(t),x'2= p21(t)x1 + p22(t)x2 + ・・・ + p2n(t)xn + g2(t),...x'n= pn1(t)x1 + pn2 (t)x2 + ・・・ + pnn(t)xn + gn(t)or using matrix notation,x' = P(t)x + g(t),P(t) is matrix of coefficients. g(t) the nonhomogeneous term of the system. If g(t) = 0 for all t ∈ I , then the system is said to be homogeneous; otherwise the system is said to be nonhomogeneous. The function g(t) is often referred to as the input, or forcing function.
6 First Order Linear Systems: General Framework (Ctd.) The system is said to have a solution on the interval I if there exists a vector x = φ(t) with n components that is differentiable at all points in the interval I and satisfies Eq. at all points in this interval.In addition to the system of differential equations, if there is an initial condition of the form x(t0) = x0, where t0 is a specified value of t in I and x0 is a given constant vector with n components.The system and the initial condition together form an initial value problem.The components of x are again referred to as state variables and the vector x = φ(t) is referred to as the state of the system at time t.
7 Linear nth Order Equations. An nth order linear differential equation in standard form is given byto obtain a unique solution, we need n initial conditionsTo transform Eq. into a system of n first order equations, we introduce the variables x1, x2, , xn defined by x1 = y, x2 = y', x3 = y'' , ..., xn = y(n−1).In Matrix
8 Applications Modeled by First Order Linear Systems We discuss several examples of applications that illustrate how higher dimensional linear systems can arise.Coupled Mass-Spring Systems.Compartment Models.Linear Control Systems.The State Variable Approach to Circuit Analysis.
9 6.2 Basic Theory of First Order Linear Systems THEOREM (Existence and Uniqueness for First Order Linear Systems).If P(t) and g(t) are continuous on an open interval I = (α, β), then there exists a unique solution x = φ(t) to the initial value problem x' = P(t)x + g(t), x(t0) = x0, where t0 is any point in I, and x0 is any constant vector with n components. Moreover the solution exists throughout the interval I.
10 THEOREM 6.2.2 (Principle of Superposition). If x1, x2, , xk are solutions of the homogeneous linear systemx' = P(t)xon the interval I = (α, β), then the linear combinationc1x1 + c2x2 + ・・・ + ckxkis also a solution of the system on I.
11 DEFINITION 6.2.3The n vector functions x1, x2, , xn are said to be linearly independent on an interval I if the only constants c1, c2, , cn such that c1x1(t) + ・・・ + cnxn(t) = 0 for all t ∈ I are c1=c2=. . . =cn = 0.If there exist constants c1, c2, , cn , not all zero, such that c1x1(t) + ・・・ + cnxn(t) = 0 is true for all t ∈ I , the vector functions are said to be linearly dependent on I.
13 DEFINITION 6.2.4Let x1, , xn be n solutions of x' = P(t)x and let X(t) be the n×n matrix whose j th column is xj (t), j = 1, , n,The Wronskian W = W[x1, , xn] of the n solutions x1, , xn is defined byW[x1, , xn](t) = det X(t).
14 THEOREM 6.2.5Let x1, , xn be solutions of x' = P(t)x on an interval I = (α, β) in which P(t) is continuous.(i) If x1, , xn are linearly independent on I, thenW[x1, , xn](t) ≠ 0 at every point in I,(ii) If x1, , xn are linearly dependent on I, thenW[x1, , xn](t) = 0 at every point in I.
18 Linear nth Order Equations. COROLLARY 6.2.8 If the functions p1(t), p2(t), , pn(t), and g(t) are continuous on the open intervalI = (α, β), then there exists exactly one solution y = φ(t) of the differential equationthat also satisfies the initial conditionsThis solution exists throughout the interval I.
20 A fundamental set of solutions and the general solution The terminology for solutions of the nth order scalar equation is identical to that used for solutions of x' = P(t)x. A set of solutions y1, , yn such that W[ y1, , yn](t0) ≠ 0 for some t0∈ (α, β) is called a fundamental set of solutions for Eq. (28).The n-parameter family represented by the linear combinationy = c1y1(t) + ・・ ・+cnyn(t),where c1, , cn are arbitrary constants, is called the general solution of Eq. (28).Corollary guarantees that each solution of Eq. (28) corresponds to some member of this n- parameter family of solutions.
21 ExampleQ: Show that y1(x)=x, y2(x)=x−1, and y3(x)=x2 constitute a fundamental set of solutions forx3 y'' + x2 y'' − 2xy' + 2y = 0 on I = (0,∞).A:1. Substituting each of y1, y2, and y3 into Eq. show that they are solutions on the given interval.2. Verify that the three functions are linearly independent on I, by computing the Wronskian3. Apply Corollary
22 6.3 Homogeneous Linear Systems with Constant Coefficients In this section, we extend the eigenvalue method to the system x' = Ax, where A is a real constant n × n matrix. As in Chapter 3, we assume solutions of the form x = eλtv, where the scalar λ and the constant n × 1 vector v are to be determined.Given a square matrix A, recall that the problem of(i) finding values of λ for which Eq. (A − λIn)v = 0 has nontrivial solution vectors v, and(ii) finding the corresponding nontrivial solutions, is known as the eigenvalue problem for A.
23 Three cases1. A has a complete set of n linearly independent eigenvectors and all of the eigenvalues of A are real,2. A has a complete set of n linearly independent eigenvectors and one or more complex conjugate pairs of eigenvalues,3. A is defective, that is, there are one or more eigenvalues of A for which the geometric multiplicity is less than the algebraic multiplicity (see Appendix A.4).
24 The Matrix A Is Nondefective With Real Eigenvalues THEOREM 6.3.1Example
25 Real and Distinct Eigenvalues. COROLLARY 6.3.2EXAMPLE
26 Symmetric Matrices.Properties of the eigenvalues and eigenvectors of a real symmetric matrix A (see Appendix A.4):1. All the eigenvalues λ1, , λn of A are real.2. A has a complete set of n real and linearly independent eigenvectors v1, , vn.Furthermore eigenvectors corresponding to different eigenvalues are orthogonal to one another and all eigenvectors belonging to the same eigenvalue can be chosen to be orthogonal to one another.
27 ExampleDiffusion on a One-Dimensional Lattice with Reflecting Boundaries.the system describing diffusion on a lattice consisting of n = 3 points is expressed in matrix notation asInsert images
28 6.4 Nondefective Matrices with Complex Eigenvalues In this section, we again consider a system of n linear homogeneous equations with constant coefficients, x' = Ax, where the coefficient matrix A is real-valued, nondefective, and has one or more complex eigenvalues.If λ = μ + iν and (λ, v) is an eigenpair of A and v = a + ib, where a and b are real constant n×1 vectors, then the vectorsx1(t) = Re u(t) = eμt (a cos νt − b sin νt)x2(t) = Im u(t) = eμt (a sin νt + b cos νt)are real-valued solutions of Eq. and are linearly independent.
29 Nondefective Matrices with Complex Eigenvalues If all of the eigenvectors of A, real and complex, are linearly independent, then a fundamental set of real solutions of x' = Ax consists of solutions of the form above associated with complex eigenvalues and solutions of the form eλjtvj associated with real eigenvalues.For example, suppose that λ1 = μ + iν, λ2 = μ − iν, and that λ3, , λn are all real and distinct. Let the corresponding eigenvectors be v1 = a + ib, v2 = a − ib, v3, , vn. Then the general solution of x' = Ax isx = c1x1(t) + c2x2(t) + c3eλ3tv3 + ・・・ + cneλnt vn,where x1(t) and x2(t) are given by above equations.We emphasize that this analysis applies only if the coefficient matrix A is real, for it is only then that complex eigenvalues and eigenvectors occur in conjugate pairs.
31 Plots of the components and A typical solution trajectory
32 Natural Frequencies and Principal Modes of Vibration. Consider again the system of two massesand three springs shown in Figure.If we assume that there are no external forces, then F1(t) = 0, F2(t) = 0 and from Section 6.1 we get the homogeneous system
33 Natural Frequencies and Principal Modes of Vibration (Ctd.) This system is said to have two degrees of freedom since there are two independent coordinates, x1 and x2, necessary to describe the motion of the system.If both masses will undergo harmonic motion at the same frequency, the frequency is called a natural frequency of the system, and the motion is called a principal mode of vibration.The principal mode of vibration corresponding to the lowest natural frequency is referred to as the first mode. The principal mode of vibration corresponding to the next higher frequency is called the second mode and so on.
34 ExampleAnswer: The general solution for arbitrary constants c1, c2, c3, and c4 is
35 The first principal mode of vibration The motions of m1 and m2 are in phase but have different amplitudes.
36 The second principal mode of vibration The motions of m1 and m2 are 180 degrees out of phase and have different amplitudes.
37 Component plotsComponent plots of the solution corresponding to the initial condition x(0) = (0, 2, 0, 0)T