Systems of Differential Equations - Math

Chapter 11. Systems of Differential Equations : Examples of Systems : Basic First-order system Methods : Structure of Linear Systems : Matrix Exponential : The Eigenanalysis Method for x = Ax : Jordan Form and Eigenanalysis : Nonhomogeneous Linear Systems : Second-order Systems : Numerical Methods for Systems Linear Systems . A linear system is a system of differential equa- tions of the form x 1 = a11 x1 + + a1n xn + f1 , x 2 = a21 x1 + + a2n xn + f2 , (1) .. x m = am1 x1 + + amn xn + fm , where = d/dt. Given are the functions aij (t) and fj (t) on some interval a < t < b. The unknowns are the functions x1 (t), .. , xn (t). The system is called homogeneous if all fj = 0, otherwise it is called non-homogeneous. Matrix Notation for Systems . A non-homogeneous system of linear Equations (1) is written as the equivalent vector-matrix system x = A(t)x + f (t), where x1 f1 a11 a1n .. x = . , f = . , A= .. xn fn am1 amn Examples of Systems 521.

522 Systems of Differential Equations Let x1(t), x2(t), x3(t) denote the amount of salt at time t in each tank. We suppose added to tank A water containing no salt. Therefore, the salt in all the tanks is eventually lost from the drains.

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