Economics Dynamics Problems 177

Economics Dynamics Problems 177 - Systems of rst-order...

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Systems of Frst-order differential equations 161 We investigated this problem in the last section. What we wish to fnd is the eigenvalues oF A and the associated eigenvectors. Return to the situation with only two variables, and let the two roots (the two eigenvalues) be real and distinct, which we shall again label as r and s . Let v r be the eigenvector associated with the root r and v s be the eigenvector associated with the root s . Then so long as r ±= s u 1 = e rt v r and u 2 = e st v s are independent solutions, while x = c 1 e rt v r + c 2 e st v s (4.22) is a general solution. Example 4.8 ±ind the general solution to the dynamic system ˙ x = x + y ˙ y =− 2 x + 4 y We can write this in matrix Form ± ˙ x ˙ y ² = ± 11 24 ²± x y ² The matrix A oF this system has already been investigated in terms oF example 4.7.
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