Differential equations

This chapter introduces methods to analyse ordinary differential equations. In this course, no other types of differential equations are discussed. Therefore, in the following chapter, all differential equations are assumed to be ordinary, i.e. only contain derivatives with respect to time. In addition, most differential equations will be autonomous, i.e. independent on time.

The differential equations can be used to describe the behavior of a system over time. Hereto, the changes in time of a system are prescribed as a function of the current state of the system.

For example, the dynamics of a mechanical system with mass $ m=1$ , spring with stiffness $ k=1$ and damping $ c=0.2$ can be described by the differential equation:

$\displaystyle m\ddot{x}+c\dot{x}+kx=0,$ (5.1)

where $ x(t)$ denotes the position of the mass. Here, the time derivatives $ \ddot{x}=\frac{d^2x(t)}{dt^2}$ and $ \dot{x}=\frac{dx(t)}{dt}$ represent the acceleration and velocity, respectively.

Another possible notation is the following form

$\displaystyle \dot{q}=Aq,$ (5.2)

where $ A=\begin{pmatrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{pmatrix}$ and the state vector $ q=\begin{pmatrix}x\\ \dot{x}\end{pmatrix}$ is chosen.

Hence, since $ q$ is a 2-dimensional vector, it is possible to depict the direction field of a differential equation in the phase plane, i.e. the plane $ x,\dot{x}$ . Hereto, depict at certain predefined coordinates the vector $ \dot{q}$ . For the differential equation shown above, this yields Figure 5.1. This figure is called the direction field, which provides insight in the behavior of the system for all initial conditions inside its domain.

Figure 5.1: Direction field

To obtain a trajectory satisfying the system equations, an initial condition should be specified. When an initial condition is specified, one can look for a trajectory with this initial condition that satisfies the differential equation. For example, in the above example, a system with initial position $ x_0=1.5,\,\dot{x}_0=0$ has the solution:

$\displaystyle x(t)=1.5 e^{-\frac{c}{2m}t}cos\left(\sqrt{\frac k m}\sqrt{1-\frac {c^2} {4km}}t\right).$ (5.3)

Plotting this function in time yields the trajectory depicted in Figure 5.2.

Figure 5.2: Trajectory in time

This trajectory can also be plotted in the phase plane, such that one obtains Figure 5.3.

Figure 5.3: Trajectory in phase plane

As can be seen in this figure, the vector field is always locally tangent to the trajectories $ q(t)$ . Therefore, these pictures can be used to visualize many different trajectories. In our example, according to the phase plane representation, all trajectories apparently encircle the origin and converge to it. Such dynamics is expected from a weakly damped system.

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Esteur 2010-03-22