Find bases for the vector spaces of solutions to the following differential equations.
Let , , and , and consider the following linear system of first order differential equations.
Which we can express as a matrix equation by
Now let denote this coefficient matrix. Let ; evidently, is in Jordan canonical form. (Computations performed by WolframAlpha.) Now is a fundamental matrix of our linear system. Reading off the first row of this matrix (and multiplying by 9), we see that , , and are solutions of our original 3rd order differential equation. Moreover, it is clear that these are linearly independent.
For the second equation, we follow a similar strategy and see that the corresponding coefficient matrix is . Let . Evidently, is in Jordan canonical form. (Computation performed by WolframAlpha.) Taking the first row of (and multiplying by -1), we see that , , , and are linearly independent solutions of the original 4th order differential equation.