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Mechanical Engineers Handbook Episode 15

Tham khảo tài liệu 'mechanical engineers handbook episode 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2. Systems of Differential Equations 829 then the real part of the solution J and the imaginary part V are suspected to be solutions of the equations L X J and L X V m 2.4.1 THE METHOD OF VARIATION OF ARBITRARY PARAMETERS THE LAGRANGE METHOD If the general solution of the corresponding homogeneous system of equations 2.21 is known and one cannot choose a particular solution of the system of equations 2.20 then the method of variation of parameters may be applied. Let X pn i CịXị be the general solution of the system 2.21 . The solution of the nonhomogeneous system 2.20 must be of the form n X t Ci t Xi 2 23 i 1 where c t are the new unknown functions. If we substitute into the nonhomogeneous equation we obtain n P C0 t Xi F i 1 This vector equation is equivalent to a system of n equations Differential Equations n P c0 t x1i f1 t i 1 . P C0 t x2i f2 t 2 24 i 1 P ci t Xni fn t i 1 All C0 t are determined from this system ci t jj t i 1 2 . n whence Ci t f ji t dt Ci i 1 2 . n The system x11 x12 x1n x21 x22 x2n X1 . . X2 . . . . Xn . . . . xn1 . . . xn2 . . xnn of particular solutions of the homogeneous system of differential equations is said to be fundamental in the interval a b if its Wronskian W t W X1 X2 . Xn x11 t x21 t x12 t x22 t x1n t x.n t 0 xn1 t xn2 t xnn t 830 Appendix Differential Equations and Systems of Differential Equations for all t 2 a b . In this case the matrix 2 X1i t X21 t Differential Equations M t xn1 t x12 t x1n t x22 t x2n t xn2 t xnn t _ 2 25 is said to be a fundamental matrix. The general solution of the homogeneous linear system of equations 2.21 is X t M t c 8 c1 9 2 26 c cn The solution of the homogeneous system dX I AX dt that satisfies the initial condition X t0 X0 is X t M t M-1 tũ X0 The system of equations 2.24 may be written in the form M t c t F t 2 27 and hence c t t M - s F s ds c 0 The general solution of the system of equations 2.16 is t M 1 s F s ds X t M t c M t 0 and the solution that satisfies X t0 X0 is X t M t M-1 t0 X0