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Matrix Methods for Linear Systems of Differential …
Matrix Methods for Linear Systems of Differential EquationsWe now present an application of Matrix Methods to Linear Systems of Differential equations . We shallfollow the development given in Chapter 9 ofFundamentals of Differential equations and BoundaryValue Problemsby Nagle, Saff, Snider, third of MatricesIf we allow the entriesaij t in ann nmatrixA t to be functions of the variablet,thenA t is amatrix function of t. Similarly if the entriesxi t of a vectorx t are functions oft,thenx t is avectorfunction of t. A matrixA t is said to becontinuous at t0if eachaij t is continuous t isdifferentiable at t0if eachaij t is differentiable att0and we writedAdt t0 A t0 aij t0 n nAlso abA t dt abaij t dtn nWe have the following differentiation formulas for matricesddt CA CdAdt,Ca constant matrixddt A B dAdt dBdtddt AB AdBdt BdAdtIn the last formula the order in which the matrices are written is important, since Matrix multiplicationneed not be Systems in Normal FormAsystemofnlinear Differential equations is innormal formif it is expressed asx t A t x t f t 1 wherex t andf t aren 1 column vectors andA t aij t n system is calledhomogeneousiff t 0; otherwise it is the1elements ofAare constants, the system is said to haveconstant note that a linearnthorder Differential equationy n t pn 1 t y n 1
Matrix Methods for Linear Systems of Differential Equations We now present an application of matrix methods to linear systems of differential equations.
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