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 for and we see that this does not work; however, 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 where is a typical eigenvector; and we see that where is a normal eigenvector defined by

Thus our fundamental set of solutions is:

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