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 $\mathbf {X} =\mathbf {u} te^{\mathbf {\lambda } t}$ 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.