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Handbook of mathematics for engineers and scienteists part 81

Handbook of mathematics for engineers and scienteists part 81. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 528 Ordinary Differential Equations 12.6. Linear Systems of Ordinary Differential Equations 12.6.1. Systems of Linear Constant-Coefficient Equations 12.6.1-1. Systems of first-order linear homogeneous equations. The general solution. 1 . In general a homogeneous linear system of constant-coefficient first-order ordinary differential equations has the form yi any 012 2 aln yn y2 021 yi O22y2 O2n yn . y n aniyi On2 y2 nnyn 12.6.1.1 where a prime stands for the derivative with respect to x. In the sequel all the coefficients aj of the system are assumed to be real numbers. The homogeneous system 12.6.1.1 has the trivial particular solution y1 y2 yn 0. Superposition principle for a homogeneous system any linear combination of particular solutions of system 12.6.1.1 is also a solution of this system. The general solution of the system of differential equations 12.6.1.1 is the sum of its n linearly independent nontrivial particular solutions multiplied by an arbitrary constant. System 12.6.1.1 can be reduced to a single homogeneous linear constant-coefficient nth-order equation see Paragraph 12.7.1-3. 2 . For brevity and clearness system 12.6.1.1 is conventionally written in vector-matrix form y ay 12.6.1.2 where y y1 y2 . yn T is the column vector of the unknowns and a aij is the matrix of the equation coefficients. The superscript T denotes the transpose of a matrix or a vector. So for example a row vector is converted into a column vector y1 y2 T f y1Y y2 The right-hand side of equation 12.6.1.2 is the product of the n X n square matrix a by the n X 1 matrix column vector y. Let yk yk1 yk2 . ykn T be linearly independent particular solutions of the homogeneous system 12.6.1.1 where k 1 2 . n the first subscript in ykm ykm x denotes the number of the solution and the second subscript indicates the component of the vector solution. Then the general solution of the homogeneous system 12.6.1.2 is expressed as y Cy C2y2 Cnyn. 12.6.1.3 A method for the construction of .