In this chapter we describe three different methods to find closed form solutions to planar constant coefficient systems of linear differential equations. In Section ?? we begin by discussing those systems of differential equations that have unique solutions to initial value problems, and these systems include linear systems. Then we show how uniqueness to initial value problems implies that the space of solutions to a constant coefficient system of linear differential equations is dimensional. Using this observation we present a direct method for solving planar linear systems in Section ??. This method extends the discussion of solutions to systems whose coefficient matrices have distinct real eigenvalues given in Section ?? to the cases of complex eigenvalues and equal real eigenvalues.

The matrix exponential is an elementary function that allows us to solve all initial value problems for constant coefficient linear systems, and this function is introduced in Section ??. In Section ?? we compute the matrix exponential for several special, but important, examples.

We compute matrix exponentials in two different ways. The first approach is based on changes of coordinates. The idea is to make the coefficient matrix of the differential equation as simple as possible; indeed we put the coefficient matrix in the form of one of the matrices whose exponential is computed in Section ??. This idea leads to the notion of similarity of matrices, which is discussed in Section ??, and leads to the second method for solving planar linear systems. Both the direct method and the method based on similarity require being able to compute the eigenvalues and the eigenvectors of the coefficient matrix.

Once these ideas have been introduced and discussed, we use the Cayley Hamilton theorem to derive a computable formula for all matrix exponentials of matrices. This formula requires knowing the eigenvalues of the coefficient matrix — but not its eigenvectors. See Section ??.

In the last section of this chapter, Section ??, we consider solutions to second order equations by reduction to first order systems.