9.2 Differential equations as Generators of Functions

The example of the distance traveled also gives an easy answer to the question: how does one find the functions that are important in Physics? How are functions defined, that describe certain situations? How does one find the relationship between variable and function value that is expressed in the function, i.e. the “character” of a special function type?

Differential equations are the parents of the functions and we will soon see, that a single differential equation, i.e. a simple relationship, creates many related children – read functions.

As discussed in chapter 5, functions describe the dependence of a quantity y on one or more other quantities, that are called variables of the function, or more correctly independent variables

For one variable, that we call t, these function shows y in its dependence from this single variable t.( We here choose t as symbol of the variable, since many examples will demonstrate a dependence on time.) The curve of the function can be visualized with a y(t) plot. Changes are described by derivatives with respect to the single variable t.

function y = y(t),variable:t1 < t < t2, slope y(t) = dy dt ;curvature y(t) = d2y dt2 = dy dt

What happens, if several variables have to be taken into account, for the example of a time dependent function of two position variables,i.e x = f(x,y,t), as we have visualized above? Here the partial differential quotients appear, that describe the change of the function value z when varying one of the variables. For the partial derivative of a function with respect to one of the variable all other independent variables are treated as constants.

z = z(x,y,t) variable:x1 < x < x2;y1 < y < y2;t1 < t < t2; z t ; z x; z y; 2z t2 ; 2z x2 ; 2z y2 ; 2z xy...

We now go back to relationships for one independent variable and consider a simple example: we know about the exponential function, that its instantaneous growth, the growth rate or slope is the exactly the same as its function value. The growth rate is identical to the first differential quotient:

fory = etwe have y = dy dt = et; y = y

This differential equation (relationship between the function and its derivatives), characterizes the nature of all growth functions (exponential functions) in a unique way. To fix a specific growth function one only requires its initial value.

If as above we only deal with the differential dependence on one variable, we refer to an ordinary differential equation. Partial differential quotients appear when there is a dependence on many variables and we refer to a partial differential equation (see below).

There is no other function that shows the same property of the derivative as the exponential function. This applies irrespective of its “amplitude”, i.e. a multiplicative factor C, because:

fory = Cetwe havey = dy dt = Cet = y y = y

In general every linear differential equation is independent of multiplicative factors.

It is quite easy to formally derive the exponential function as solution from the knowledge of the differential equation using elementary integrals.

y = y dy dt = y; the solution method of choiceis the seperation of variables  dy y = dt, integration of left hand side y(0)y1 ydy = lny - lny0mity(0) = y0, integration of right hand side0tdt = t - 0 lny = t + lny0;y = et+ln y0 = y0et

The basis exponential function y = et is the result for the initial valuey0 = 1. From the last equation one can see, that the multiplication with the initial value yields the same function as the translation along the t-axis by the logarithm of the initial value.

Figure 9.1: Family of solutions y = Aex of y = y. The parameter of the curves is the initial value A = y(0) (intersection with the ordinate), which takes on the integer values from -20 to +20.

For different initial values the differential equation describes a family of exponential functions, that are distinguished by a multiplicative factor. The diagram in Fig.9.1 shows this family of curves for positive and negative initial values between -20 and 20 with step width 1.

In general an ordinary differential equation is defined by a functional relationship between function, its derivatives and the variable t

general F(t,y,y,y,.....,y(n)) = 0, explicit y(n) = f((t,y,y,y,...y(n-1)).

A differential equation is called explicit if the highest derivative can be expressed as function of the lower derivatives.

The above equation for the exponential function is an ordinary explicit linear differential equation of first order. The equation is

These criteria are important for finding an analytical solution and are also important for the numerical solution with limited computational power. In the case of explicit equations and important numerical methods, only already calculated data enter the procedure to calculate a solution step by step. For implicit equations (an exotic example is ycosy + xy = 0) one has to solve for every step of the calculation an equation that already contains the results for the next step already. This is in general not possible in closed form, bot only through iteration. With sufficient computational capacity, this dilemma loses however its importance. We have already shown above, how to solve complicated equations with iterative methods.

One could of course also describe the exponential function via a differential equation of second, or even higher order, because we have

y = ex;y = ex;y = ex...... for exampley = y

If one asks, which functions satisfy this differential equation of second order one realizes, maybe initially surprisingly, that it is not satisfied only by the simple exponential function, but in addition by a multitude of functions that are related to it.

Indeed we have with A and B as constants:

Fory = Aet y = Aet y = Aet y = y(= y) For y = Ae-t y = -Ae-t y = Ae-t y = y(y) For y = Aet - Be-t y = Aet + Be-t y = Aet - Be-t y = y(y) in particular forA = B = 1 2 : y = cosh(t) = et+e-t 2 y = et-e-t 2 y = et+e-t 2 y = y(y) y = sinh(t) = et-e-t 2 y = et+e-t 2 y = et-e-t 2 y = y(y)

In addition to the simple exponential function with a positive exponent this also includes exponential functions with a negative exponent and also all linear combinations of these two components, of which we have formulated cosht and sinht at the end of the list.

In the following diagram Fig.9.2 , which is not active, we show the families of the functions described above. First the family of exponential functions with positive and negative exponents and initial values is shown.

Figure 9.2: Family of solutions y = Aex and y = Ae-x of y = y for different initial values of A = y(0). A covers all integer values from -20 to 20 .

Fig.9.3 then shows the hyperbolic functions that are either symmetric or antisymmetric to x = 0 and are determined by a single initial value A.

Asinh(x) = Aex - e-x 2 ;Acosh(x) = Aex + e-x 2

The function y(x) = Asinhx goes through (0,0) and y(x) = Acoshx goes through (0,A) with initial value y(0) = A. A changes in integer steps from -20 to +20.

Figure 9.3: Family of solutions of hyperbolic trig functions.

Finally Fig.9.4 shows the general solutions with two parameters A and B:

Aex - Be-x;Aex + Be-x with A  = 1,2,3...10  and B  =  1, 5, 10 

The choice of the parameters A and B, including their signs, determines which individual function from the abundance of functions, that satisfy the differential equation y = y, is realized. One obtains all functions, for which the curvature has the same sign and absolute value as the function value.

In this simple case it is immediately obvious, that one could instead of the differential equation of second order also two differential equations of first order:

y = y A y = y B

Figure 9.4: Two-parameter family of solutions of y = y. The vales of A and B determine for example the two initial values y(0) = A + B and y(0) = A - B.

In general one can reduce a ordinary differential equation of n-th order to a system of n equation of first order:

y(n) = f(y,y,y...,y(n-1),t) 1)y = y1 2)y1 = y2 3)y2 = y3 4.)y3 = y4 ... yn-1 = f(y,y1,y2,y3,...,yn-1,t) Since the differential equation is explicit, f(...) does not depend on yn 

The differential equation of n-th order has n parameters of initial values, that are in general different from each other. By fixing specific values for the parameters one selects from all functions, that satisfy the differential equations a special one. The physical solution is thus obtained from the differential equations and the initial values.

The beauty and descriptive power of mathematics with its application in Physics shines in a very special manner for the differential equations. One single, formally quite simple relationship can include a multitude of solution possibilities, out of which the selection of a few parameters picks the special solution. Understanding the relationships of the differential equations therefore is much more important than the knowledge of a large number of formulas for limited problem areas.

The differential equation y = -y describes all phenomena in connection with undamped, sine-shaped oscillations. A factor a (y = -ay) does not change anything fundamental, since scaling the time with t = at transforms the differential equation back to y = -y. What is the visual meaning of this equation ? The curvature y is equal to the negative function value. This means for a large positive function value the curvature will finally reduce the function value, and for a negative value the absolute value finally decreases. However, this is exactly the hallmark of a periodic oscillation: the values do not go beyond a maximum or minimum value, but are always led from the one to the other. For damped oscillations or those whose amplitude increases with time, one uses the more general law y = ay- by. This says, that for a > 0 the curvature increases with time, thus the oscillation grows, while for a < 0 the curvature decreases, thus the oscillation decays.

Other classes of phenomena can be described via different classes of differential equations. For example. the Newton equation of motion md2r dt2 = F(r) governs the huge class of all possible movements of a mass under the influence of a given force field F(r). This includes for example planet movements. The resulting mechanical movements depend on the form of the force field and especially on the initial values.

Thus differential equations are the condensed information to classify a far range of physical phenomena. They also have a wide range of applicability as well as high aesthetic appeal and provide order in the plethora of natural phenomena.