常微分方程第8章.ppt

8 systems of linear first-order differential equations or ? If is a fundamental matrix of the system then Variation of parameters Replace the matrix of constants C in (1) by a column matrix of functions is a particular solution of the nonhomogeneous system By the product rule, have Substituting (4) and (6) into (5) gives We have Example 1 Find the general solution of the nonhomogeneous system On the interval . Solution we first solve the homogeneous system The characteristic equation of the coefficient matrix is The eigenvalues corresponding to and are, respectively, The solution vectors of the system are then hence From the formula we obtain Hence, the general solution is Initial-value problem The general solution of (5) can be written as The form is useful in solving (5) subject to an initial condition . 8.4 matrix exponential The simple linear first-order differential equation , where is a constant, has the general solution The matrix exponential is a solution of the system Homogeneous systems We define a matrix exponential so that is solution of the homogeneous system Here A is an matrix of constants, and C is an column matrix of arbitrary constants. The power series(幂级数) representation of the scalar exponential function The series in(2) converges for all t. DEFINITION 8.4 Matrix Exponential For any matrix A, The series given in(3) converges to an matrix for value of t. and Derivative of We have known and we can justify We can get Hence, is a solution of for every vector C of constants. EXAMPLE 4 Repeated Eigenvalues Find general solution of the system given in (10). Soluti