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Unformatted text preview: Linear Systems of Differential Equations A first order linear n-dimensional system of differential equations takes the form Y ( t ) = A ( t ) Y ( t ) + B ( t ) , or, in expanded form, y 1 ( t ) y 2 ( t ) . . . y n ( t ) = a 11 ( t ) a 12 ( t ) a 1 n ( t ) a 21 ( t ) a 22 ( t ) a 2 n ( t ) . . . . . . . . . a n 1 ( t ) a n 2 ( t ) a nn ( t ) y 1 ( t ) y 2 ( t ) . . . y n ( t ) + b 1 ( t ) b 2 ( t ) . . . b n ( t ) . As usual, we define a solution of this system to be a differentiable n-vector function Y ( t ) which reduces the above to an identity upon substitution. The system is homogeneous if B ( t ) 0 (the zero vector), inhomogeneous otherwise. In our discussion we will assume that the functions a kj ( t ) forming the entries of the matrix A ( t ) and the functions b k ( t ) forming the com- ponents of the vector function B ( t ) are (at least) piecewise continuous functions of the independent variable t ; most examples involve contin- uous functions of t . Example 1 The system of equations y 1 ( t ) y 2 ( t ) ! = 1 t y 1 ( t ) + y 2 ( t ) 2 t y 2 ( t ) constitutes a 2 dimensional linear first order homogeneous system of differential equations, 0 < t < . If we change the system to y 1 ( t ) y 2 ( t ) ! = 1 t y 1 ( t ) + y 2 ( t ) + t 2 t y 2 ( t ) + t 2 1 we have a 2 dimensional linear first order inhomogeneous system of differential equations. Here we have A ( t ) = 1 t 1 2 t ! , B ( t ) = t t 2 . The general solution of a linear homogeneous system Y ( t ) = A ( t ) Y ( t ) takes the form Y ( t, c 1 , c 2 , ..., c n ) = c 1 Y 1 ( t ) + c 2 Y 2 ( t ) + + c n Y n ( t ) , where in this formula the Y k ( t ) , k = 1 , 2 , ..., n , are n-vector solutions of the system; thus Y k ( t ) = y 1 k ( t ) y 2 k ( t ) . . . y nk ( t ) . Further, these solutions should constitute a fundamental set of n- vector solutions, by which we mean that, given any value of t in an interval ( a, b ) in which the system satisfies our basic assumptions (continuity, etc.), and given an initial vector Y = y 10 y 20 ....
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