Ordinary differential equations

Matrix differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a first-order matrix ordinary differential equation is where is an vector of functions of an underlying variable , is the vector of first derivatives of these functions, and is an matrix of coefficients. In the case where is constant and has n linearly independent eigenvectors, this differential equation has the following general solution, where λ1, λ2, …, λn are the eigenvalues of A; u1, u2, …, un are the respective eigenvectors of A; and c1, c2, …, cn are constants. More generally, if commutes with its integral then the general solution to the differential equation is where is an constant vector. By use of the Cayley–Hamilton theorem and Vandermonde-type matrices, this formal matrix exponential solution may be reduced to a simple form. Below, this solution is displayed in terms of Putzer's algorithm. (Wikipedia).

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From playlist Matrix Equations

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From playlist Systems of Differential Equations

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