Ordinary 2nd Order Linear Differential Equations

We consider a particle of mass m that is moving along a straight line in x–direction. At time t, its coordinate is x = x(t). It is attached to springs with spring constant k > 0 so that there is a ‘restoring’ force {f}_{r}(x) = −kx acting on the particle. At x = 0, the mass is in equilibrium and no force is acting. In addition, there is a friction force {f}_{f}(v) = −γv acting on the particle which is proportional (with friction constant γ > 0) to its velocity v =\dot{ x}(t), and an external force {f}_{e}(x) that could have its origin in, e.g., some crazy experimentalist fiercly forcing the mass to follow her hand.

Newton’s law states that m\ddot{x}(t) equals the sum {f}_{r}(x) + {f}_{f}(x) + {f}_{e}(x) of all forces on the particle, i.e.

\begin{eqnarray} m\ddot{x}(t)& =& −kx − γ\dot{x}(t) + {f}_{e}(x) ⇔ %& \\ \ddot{x}(t) + {γ\over m}\dot{x}(t) + {k\over m}x(t)& =& {1\over m}{f}_{e}(x),\quad k > 0,γ > 0. %&(2.1) \\ \end{eqnarray}

To find the position x of the particle at time t, i.e. the function x(t), we have to solve the differential equation of the forced, damped linear harmonic oscillator, Eq. (2.1). Learn this standard form of the forced damped harmonic oscillator by heart and it will save you from much misery in the future.

CHECK: to which forces do the terms ‘forced’, ‘damped’, and ‘harmonic’ refer ?

Is this a well–defined task? No, in order to know x(t) at all times later than, say, t = 0, we must specify the initial conditions, i.e. the initial position of the particle x(t = 0) and its initial velocity \dot{x}(t = 0).

Eq. (2.1)is called 2nd order differential equation because the highest derivative appearing is a second derivative. Because Newton’s law (for a general force) leads to second derivatives (acceleration term!), 2nd order differential equations belong to the most important differential equations in physics.

Eq. (2.1) is called linear because we don’t have terms like \ddot{{x}}^{2}(t) or {x}^{4}(t). In general and in more complicated cases (e.g., motion in three dimensions), such terms can leads to chaos. The study of differential equations therefore is of paramount importance in order to understand chaos.

Eq. (2.1) is called Ordinary because the desired function x is a function of one variable (t) only and not more than one variable, in which case differential equations are called partial differential equations.

2.1.2 Definitions

In the mathematic literature, people sometimes don’t care about the physical background of equations and introduce other notations. In the following, instead of x(t), \dot{x}(t) etc we discuss differential equations for functions y(x) of one variable x, with y'(x) denoting the first and y''(x) the second derivative, respectively.

A 2nd order inhomogeneous linear differential equation for the function y(x) has the form

\begin{eqnarray} y''(x) + p(x)y'(x) + q(x)y(x) = f(x),& & %&(2.2) \\ \end{eqnarray}

where p(x), q(x), and f(x) are known functions of x and y(x) is the function one would like to calculate.

In general, there is no method to obtain a solution y(x) of Eq. (2.2 that could be written down in a simple form, such as y(x) =\mathop{ sin}\nolimits (x) etc.

A 2nd order homogeneous linear differential equation for the function y(x) has the form

\begin{eqnarray} y''(x) + p(x)y'(x) + q(x)y(x) = 0,& & %&(2.3) \\ \end{eqnarray}

A 2nd order inhomogeneous linear differential equation for the function y(x) with constant coefficients has the form

\begin{eqnarray} y''(x) + py'(x) + qy(x) = f(x),& & %&(2.4) \\ \end{eqnarray}

where p and q are real numbers, f(x) is a known function of x, and y(x) is the function one would like to calculate.

A 2nd order homogeneous linear differential equation for the function y(x) with constant coefficients has the form

\begin{eqnarray} y''(x) + py'(x) + qy(x) = 0,& & %&(2.5) \\ \end{eqnarray}

where p and q are real numbers, and y(x) is the function one would like to calculate.

Initial Value Problem for 2nd order differential equation for a function y(x): To solve the initial value problem for a 2nd order differential equation for a function y(x) means to solve y(x) for the specific, given initial conditions

\begin{eqnarray} y(x = {x}_{0}) = {y}_{0},\quad y'(x = {x}_{0}) = {y'}_{0}.& & %&(2.6) \\ \end{eqnarray}

In the example of our harmonic oscillator this means that we start the motion at t = {t}_{0} = 0 at the initial position x({t}_{0}) = {x}_{0} with the initial velocity \dot{x}({t}_{0}) =\dot{ {x}}_{0}.

2.1.3 How to Solve Them

In general, there is no recipe or general method of how to solve a given differential equation. In this lecture, we only discuss the 2nd order inhomogeneous linear differential equation for the function y(x) with constant coefficients, for which there is a general method. ‘Differential Equations’ is a difficult topic, and still today a research subject in mathematics. Generations of people have tried to solve differential equations by finding new exact solutions, developing approximation techniques etc. For example, a big problem in Einstein’s theory of gravitation is that the fundamental (partial) differential equations are known, but only very few exact solutions are known. This is still a hot topic today.

EXAMPLE: a particle of mass m under a constant external force {f}_{e}(x) = {f}_{e} that does not depend on x. We have

\begin{eqnarray} \ddot{x}(t)& =& {1\over m}{f}_{e} ⇝\dot{ x}(t) = {1\over m}{f}_{e}t +\dot{ x}(0) ⇝ %& \\ x(t)& =& {1\over 2m}{f}_{e}{t}^{2} +\dot{ x}(0)t + x(0). %&(2.7) \\ \end{eqnarray}

Here, the values x(0) and \dot{x}(t = 0) determine the initial condition at t = 0.

CHECK: go back to Pisa (Galilei) and establish the relation between this equation and the experiment of a freely falling mass m. In a ‘Gedankenexperiment’ (thought experiment), change the initial conditions \dot{x}(t = 0) and x(0) and discuss what changes then. What does a positive or a negative {f}_{e} mean?

2.1 Ordinary 2nd Order Linear Differential Equations

2.1.1 Origin of Differential Equations: the Harmonic Oscillator as an Example

2.1.2 Definitions

2.1.3 How to Solve Them