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 $$\mathbf{X} = \mathbf{u} t e^{\mathbf{\lambda}t}$$ for $$\mathbf{X}' = \mathbf{A} \mathbf{X}$$ and we see that this does not work; however, $$\mathbf{X} = (\mathbf{B} t + \mathbf{C}) e^{\mathbf{\lambda}t}$$ 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 $$\mathbf{B} = \mathbf{u}$$ where $$\mathbf{u}$$ is a typical eigenvector; and we see that $$\mathbf{C} = \mathbf{n}$$ where $$\mathbf{n}$$ is a normal eigenvector defined by $$(\mathbf{A} - \lambda \mathbf{I})\mathbf{C} = \mathbf{B}$$

Thus our fundamental set of solutions is: $$\{\mathbf{u}te^{\lambda t} + \mathbf{n}e^{\lambda t};\mathbf{u}e^{\lambda t}\}$$

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