Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Real, Distinct Eigenvalues Method

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If the eigenvalues for the characteristic equation are real and distinct, mathematically, nothing is really wrong. Thus, by our guess and the existence and uniqueness theorem, for an n-size square matrix, the solution set is determined by:

\mathbf{X} = \{ e^{\lambda_1 \cdot t} ; e^{\lambda_2 \cdot t} ; ... ; e^{\lambda_{n-1} \cdot t} ; e^{\lambda_n \cdot t} \}

Then since the linear combination of two solutions is also a solution (which can be verified directly from the structure of the problem), we can form the general solution as such:

\mathbf{X} = c_1 e^{\lambda_1 \cdot t} + c_2 e^{\lambda_2 \cdot t} + ... + c_{n-1} e^{\lambda_{n-1} \cdot t} + c_n e^{\lambda_n \cdot t}

What's interesting is when the eigenvalues are not so simple.