Systems of Frst-order differential equations161We investigated this problem in the last section. What we wish to fnd is theeigenvalues oFAand the associated eigenvectors. Return to the situation with onlytwo variables, and let the two roots (the two eigenvalues) be real and distinct,which we shall again label asrands. Letvrbe the eigenvector associated with therootrandvsbe the eigenvector associated with the roots. Then so long asr±=su1=ertvrandu2=estvsare independent solutions, whilex=c1ertvr+c2estvs(4.22)is a general solution.Example 4.8±ind the general solution to the dynamic system˙x=x+y˙y=−2x+4yWe can write this in matrix Form±˙x˙y²=±11−24²±xy²The matrixAoF this system has already been investigated in terms oF example 4.7.
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Eigenvalue, eigenvector and eigenspace, general solution, c2 est vs