Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Repeated Eigenvalue Method

When the eigenvalue is repeated we have a similar problem as in normal differential equations when a root is repeated, we get the same solution repeated, which isn't linearly independent, and which suggest there is a different solution. Because the case is very similar to normal differential equations, let us try $𝐗=𝐮t{e}^{\mathit{\lambda }t}$ for ${𝐗}^{\prime }=𝐀𝐗$ and we see that this does not work; however, $𝐗=(𝐁t+𝐂)e^{\mathit{\lambda }t}$ DOES work (For the observant reader, this gives a hint to the changes in the Method of Undetermined Coefficients as compared to differential equations without linear algebra).

In fact if we use this we see that $𝐁=𝐮$ where $𝐮$ is a typical eigenvector; and we see that $𝐂=𝐧$ where $𝐧$ is a normal eigenvector defined by $(𝐀-\lambda 𝐈)𝐂=𝐁$

Thus our fundamental set of solutions is: $\{𝐮t{e}^{\lambda t}+𝐧e^{\lambda t};𝐮e^{\lambda t}\}$

Using the same process of derivation, higher-order problems can be solved similarly.