Linear Differential Equations

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Primarily our discussions of differential equations have focused on two issues: generalizations of solutions of the differential equation $\dot {x}=\lambda x$ to systems of equations and the qualitative interpretation of numerically obtained phase portraits for autonomous nonlinear systems. Except for finding closed form solutions of systems of equations $\dot {X}=AX$ (in the plane, Chapter ??, or in Jordan normal form, Section ??), we have solved in closed form virtually no other differential equation. In this chapter and the next two we focus on finding closed form solutions to a variety of differential equations. This chapter and the next discuss constant coefficient linear equations, both homogeneous and inhomogeneous, while Chapter ?? discusses nonconstant coefficient and nonlinear equations.

The system of differential equations $\dot {X}=AX$ is a constant coefficient, first order, homogeneous, linear system of ordinary differential equations. We begin this chapter (Section ??) by discussing how to solve the systems $\dot {X}=AX$ in the given coordinates (rather than by first transforming $A$ to Jordan normal form, as we did in Section ??). For example, we present a formula for computing $e^{tA}$ for any matrix $A$ . Later we solve some linear equations that are neither homogeneous nor first order.

In Section ?? we discuss how to solve higher order linear differential equations by reducing them to first order systems. This section generalizes the reduction of second order equations to planar systems described in Section ??. We will see that generally it is easier to solve higher order equations in closed form than to solve first order systems. Section ?? discusses the solution of an inhomogeneous higher order equation by undetermined coefficients. The method of undetermined coefficients is most easily understood in the language of linear differential operators, and this language is introduced in Section ??.

The chapter ends with a discussion of resonance in Section ??. Here we use explicit closed form solutions found by the method of undetermined coefficients to understand a physically motivated phenomenon that occurs in solutions to forced second order equations.