Boyce/DiPrima/Meade 11th ed, Ch 7.1: Introduction to Systems of First Order Linear Equations Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • A system of simultaneous first order ordinary differential equations has the general form x1 F1(t, x1, x2 ,xn ) x2 F2 (t, x1, x2 ,xn ) xn Fn (t, x1, x2 ,xn ) where each xk is a function of t. If each Fk is a linear function of x1, x2, …, xn, then the system of equations is said to be linear, otherwise it is nonlinear. • Systems of higher order differential equations can similarly be defined. Example 1 • The motion of a certain spring-mass system from Section 3.7 was described by the differential equation 1 u¢¢(t)+ u¢(t)+ u(t) = 0 8 • This second order equation can be converted into a system of first order equations by letting x1 = u and x2 = u'. Thus x1¢ = x2 1 x2¢ + x2 + x1 = 0 or 8 x1¢ = x2 1 x¢ = -x - x 2 1 8 2 Nth Order ODEs and Linear 1st Order Systems • The method illustrated in the previous example can be used to transform an arbitrary nth order equation y(n) Ft, y, y, y,, y(n1) into a system of n first order equations, first by defining (n1) x1 y, x2 y, x3 y, , xn y Then x1 x2 x2 x3 xn 1 xn xn F(t, x1, x2 ,xn ) Solutions of First Order Systems • A system of simultaneous first order ordinary differential equations has the general form x1 F1(t, x1, x2 ,xn ) xn Fn (t, x1, x2 ,xn ). It has a solution on I : a < t < b if there exists n functions x1 1(t), x2 2 (t),, xn n (t) that are differentiable on I and satisfy the system of equations at all points t in I. • Initial conditions may also be prescribed to give an IVP: 0 0 0 x1(t0 ) x1 , x2 (t0 ) x2 ,, xn (t0 ) xn Theorem 7.1.1 • Suppose F1,…, Fn and ¶F1 / ¶x1,...,¶F1 / ¶xn,...,¶Fn / ¶x1,...,¶Fn / ¶xn x F (t, x , x ,x ) are continuous1 1 in the1 region2 n R of t x1 x2…xn-space defined by x2 F2 (t, x1, x2 ,xn ) a < t < b , a 1 < x 1 < b 1 , ..., a n < x n < b n and let the point 0 0 0 t 0 , x 1 , x 2 , , x n be contained in R. Then in some interval xn Fn (t, x1, x2 ,xn ) (t0 – h, t0 + h) there exists a unique solution x1 1(t), x2 2 (t),, xn n (t) that satisfies the IVP. Linear Systems • If each Fk is a linear function of x1, x2, …, xn, then the system of equations has the general form x1 p11(t)x1 p12(t)x2 p1n (t)xn g1(t) x2 p21(t)x1 p22(t)x2 p2n (t)xn g2 (t) xn pn1(t)x1 pn2 (t)x2 pnn(t)xn gn (t) • If each of the gk(t) is zero on I, then the system is homogeneous, otherwise it is nonhomogeneous. Theorem 7.1.2 • Suppose p11, p12,…, pnn, g1,…, gn are continuous on an interval with t0 in I, and let 0 0 0 x1 , x2 ,, xn x1 p11(t)x1 p12(t)x2 p1n (t)xn g1(t) prescribex p the(t) xinitial p (conditions.t)x p Then(t)x there g (existst) a unique solution2 21 1 22 2 2n n 2 I :a < t < b xn xpn1(t)(xt1),xpn2 (t)x(2t),,x pnn(t)(xtn) gn (t) that satisfies1 1 the IVP,2 and2 existsn throughoutn I. Boyce/DiPrima/Meade 11th ed, Ch 7.2: Review of Matrices Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • For theoretical and computational reasons, we review results of matrix theory in this section and the next. • A matrix A is an m x n rectangular array of elements, arranged in m rows and n columns, denoted a a a 11 12 1n a21 a22 a2n A a i j am1 am2 amn • Some examples of 2 x 2 matrices are given below: 1 2 1 3 1 3 2i A , B , C 3 4 2 4 4 5i 6 7i Transpose T • The transpose of A = (aij) is A = (aji). a a a a a a 11 12 1n 11 21 m1 a21 a22 a2n a12 a22 am2 A AT am1 am2 amn a1n a2n amn • For example, 1 4 1 2 T 1 3 1 2 3 T A A , B B 2 5 3 4 2 4 4 5 6 3 6 Conjugate • The conjugate of A = (aij) is A = (aij). a a a a a a 11 12 1n 11 12 1n a21 a22 a2n a21 a22 a2n A A am1 am2 amn am1 am2 amn • For example, 1 2 3i 1 2 3i A A 3 4i 4 3 4i 4 Adjoint • The adjoint of A is AT , and is denoted by A* a a a a a a 11 12 1n 11 21 m1 a21 a22 a2n a12 a22 am2 A A* am1 am2 amn a1n a2n amn • For example, 1 2 3i * 1 3 4i A A 3 4i 4 2 3i 4 Square Matrices • A square matrix A has the same number of rows and columns. That is, A is n x n. In this case, A is said to have order n. a a a 11 12 1n a21 a22 a2n A an1 an2 ann • For example, 1 2 3 1 2 A , B 4 5 6 3 4 7 8 9 Vectors • A column vector x is an n x 1 matrix. For example, 1 x 2 3 • A row vector x is a 1 x n matrix. For example, y 1 2 3 • Note here that y = xT, and that in general, if x is a column vector x, then xT is a row vector. The Zero Matrix • The zero matrix is defined to be 0 = (0), whose dimensions depend on the context. For example, 0 0 0 0 0 0 0 0 , 0 , 0 0 0, 0 0 0 0 0 0 0 Matrix Equality • Two matrices A = (aij) and B = (bij) are equal if aij = bij for all i and j. For example, 1 2 1 2 A , B A B 3 4 3 4 Matrix – Scalar Multiplication • The product of a matrix A = (aij) and a constant k is defined to be kA = (kaij). For example, 1 2 3 5 10 15 A 5A 4 5 6 20 25 30 Matrix Addition and Subtraction • The sum of two m x n matrices A = (aij) and B = (bij) is defined to be A + B = (aij + bij). For example, 1 2 5 6 6 8 A , B A B 3 4 7 8 10 12 • The difference of two m x n matrices A = (aij) and B = (bij) is defined to be A - B = (aij - bij). For example, 1 2 5 6 4 4 A , B A B 3 4 7 8 4 4 Matrix Multiplication • The product of an m x n matrix A = (aij) and an n x r matrix B = (bij) is defined to be the matrix C = (cij), where n cij aikbkj k1 • Examples (note AB does not necessarily equal BA): 1 2 1 3 1 4 3 8 5 11 A , B AB 3 4 2 4 3 8 9 16 11 25 1 9 2 12 10 14 BA 2 12 4 16 14 20 3 0 1 2 3 3 2 0 0 4 3 5 1 C , D 1 2 CD 4 5 6 12 5 0 0 10 6 17 4 0 1 Example 1: Matrix Multiplication • To illustrate matrix multiplication and show that it is not commutative, consider the following matrices: 1 2 1 2 1 1 A 0 2 1, B 1 1 0 2 1 1 2 1 1 • From the definition of matrix multiplication we have: æ 2 - 2 + 2 1+ 2 -1 -1+1 ö æ 2 2 0 ö AB = ç 2 - 2 -2 +1 -1 ÷ = ç 0 -1 -1 ÷ ç ÷ ç ÷ è 4 +1+ 2 2 -1-1 -2 +1 ø è 7 0 -1 ø æ 2 - 2 -4 + 2 -1 2 -1-1 ö æ 0 -3 0 ö BA = ç 1 -2 - 2 1+1 ÷ = ç 1 -4 2 ÷ ¹ AB ç ÷ ç ÷ è 2 + 2 -4 - 2 +1 2 +1+1 ø è 4 -5 4 ø Vector Multiplication • The dot product of two n x 1 vectors x & y is defined as n T x y xi y j k1 • The inner product of two n x 1 vectors x & y is defined as n T x,y x y xi y j k1 • Example: 1 1 T x 2 , y 2 3i x y (1)(1) (2)(2 3i) (3i)(5 5i) 12 9i 3i 5 5i x,y xT y (1)(1) (2)(2 3i) (3i)(5 5i) 18 21i Vector Length • The length of an n x 1 vector x is defined as n 1/ 2 n 1/ 2 1/ 2 2 x x,x xk xk | xk | k1 k1 • Note here that we have used the fact that if x = a + bi, then x x a bia bi a2 b2 x 2 • Example: 1 1/ 2 x 2 x x,x (1)(1) (2)(2) (3 4i)(3 4i) 3 4i 1 4 9 16 30 Orthogonality • Two n x 1 vectors x & y are orthogonal if (x,y) = 0.