DIFFERENTIAL EQUATIONS. If we have given the re lation y = sin x and differentiate twice with respect to x, we find that y"+y = o. This latter equation is an ordinary differential equation. In general, an equation involving derivatives of y with respect to x, together possibly with x and y, is called an ordinary differential equation.
In general, we say that a function y of the independent variable x is a solution of a differential equation if the two sides of the equation are identically equal—that is, equal for all values of x when for y and its various derivatives in the equation there are substituted the given function and its corresponding derivatives. It is not evident that a differential equation selected at random has a solution, but it will be shown later that under certain rather broad conditions solutions do exist. It is however evident in dependently of this that many of the simpler and more common differential equations, such as we shall discuss in the first part of this article, have solutions, as these solutions either can be seen directly or can be found by more or less ingenious devices.
Many problems might be cited from physics and chemistry to show how the statement of a natural law gives us a differential equation and how the solution of this equation furnishes infor mation concerning natural phenomena.
An ordinary differential equation is said to be of order n if it contains the nth derivative of y with respect to x, but no de rivative of higher order.
(I) Variables Separable.—If (I) can be written in such a way that one member contains only x and its differential, the equa tion can be solved by two integrations, or quadratures, as they are tegration we get log y = log x+log c, or y = cx. Since we can give to c any constant value, we have here a whole family of loci which are all straight lines through the origin. The fact that there is an unlimited number of these loci is in agreement with the theo rem to be established in the second part of this article that there is a solution passing through an arbitrary point of the plane.
(2) Exact Equations.—When (I) is written in the form Mdx+Ndy=o, (3) where M and N are functions of x and y, it may be that the left member is the exact differential of some function of x and y, as 4)(x , y). When this is the case the equation is said to be exact. The solution is then obviously given by the equation 4(x , y) = c. For example, the left member of the equation is the differential of xy. Hence the solution is given by the equa tion xy = c. The curves of this family are equilateral hyperbolas (see HYPERBOLA). On the other hand, the left member of the equation is not an exact differential. But if we divide through by we get ydx—xdy 2 = y o, and the left member of this equation is the differential of x y Hence = c gives the solution. A factor, such as in this case, y y which when introduced into equation (3) makes the left member an exact differential is called an integrating factor. Every equa tion of the form (3) has an unlimited number of integrating factors. The actual determination of such factors is, however, in most cases a difficult matter. In a few simple cases they can be determined by inspection or by the application of simple rules.
(3) Homogeneous Equations.—The function f(x, y) is said to be homogeneous of degree m if the equation f (u, v) = tmf (x, y) is identically satisfied when we put u = tx and v = ty. If now equation (I), f(x , y) is homogeneous of degree zero and we take This tells us that f(x, y) can be expressed as a function alone, and suggests that we write (I) in terms of a new variable v connected with x and y by the relation v= x , or y= vx.
In order to do this we must know what dy is in terms of v and x their differentials. We find by differentiation that dy = vdx+xdv.
dv dx Hence (I) becomes vdx -}- xdv = f (i , v) dx, or f(I v) — v = x ' and the variables are separated.
(4) Linear Equations.—Equations of the form dy +P(x)y = Q(x) (4) dx are linear with respect to y and dy • They are therefore called dx linear equations. They are also of the first order. We shall later discuss linear equations of higher orders.
We could solve (4) if we knew an integrating factor. If v is such a factor v(x)y'+P(x)v(x)y is the derivative with respect to x of some function. Since this expression contains the term vy', we are led to consider whether v can be so determined that vy'+Pvy shall be the derivative of vy. In order for this to be the case we must have Since any integrating factor will answer our purpose, we can put c =o.
If we apply this factor eJP(x)dx to (4) we get efP(x)dx y = Qe Hence e ff'(x)dx y= f or (5) Equations of Order Higher Than the First.—We mention first certain equations of the second order whose solution can be made to depend upon the solution of equations of the first order— for example, the equation d2s d (s), ds dvwhich occurs in mechanics. If we put v = d tt ' we have d t = f or v dv = f (s). This is of the first order and the variables can be ds separated. Having determined v, we can find s by a quadrature. As a second example, consider the differential equation of the catenary, Its solution can be reduced to the solution of an equation of the first order by putting dv = p.
dx The equations that are linear and homogeneous in y and its derivatives, and have constant coefficients, form another simple class under this head. They are of the form ... (5) If the right member is a function of x, instead of o, we have the non-homogeneous equation with constant coefficients + ... + pn-iy'+pny =f(x) • Equations of this form are in general much more difficult to solve than those of form (5). We shall therefore consider .first those of form (5).
In case n= 1, we have Poy'+Piy = o, and we know from the discussion of equation (4), since P is here a constant and Q = o, that y= Cerx, where r is a constant, is a solution. We are therefore led to inquire whether y = erx is a solution of (5) for a properly chosen constant value of r. If we substitute this function for y in (5), the left member becomes rn+ + . . . d- Pn-ir+Pn) .
This equals zero if, and only if, r is a root of the equation Porn+Pirn-1+ ... +pn-ir = 0. (7) This equation is called the auxiliary equation of (5). The manner of its formation from (5) is obvious.
If (7) has n distinct roots, • • • , (5) has the linearly independent solutions yi = y2 = ere x ' • • . , = ern x.
Now it is a property of linear, homogeneous differential equations that the sum of two solutions is also a solution, and that the product of a solution by any constant is a solution. Hence where
• • • ,
are arbitrary constants, is a solution of (5). Moreover any solution of (5) can be obtained from (8) by as signing to the c's proper values, as will appear from the theoretical discussion to be given later. For this reason (8) is called the general solution of (5). But if equation (7) has multiple roots we cannot get the general solution of (5) in this way, although we can get certain particular solutions. If r is a k-fold root of (7) and we put y = xierx, where i is any positive integer less than k, we shall get from the left member of (5) err [f ")
+i2x2f(i-r) (r) + ... +x`f (r) ]• i(i—
• • - (i—j+I)Now i, = • 1\ow this is zero since r is a k-fold root of (7) and i
Since we now know how to solve the homogeneous equation, we need to consider only the problem of finding a particular solution of (6). Various special devices for finding this particular solution are given in the textbooks. They depend upon special forms of the function f (Y) . A more general method due to La grange (q.v.) is based on the idea of taking the complementary function and replacing the n arbitrary constants in it by such functions of x as will give us a solution of (6). It turns out that this can be done in a variety of ways, but the details cannot be given here.
The problem of solution is still more difficult if the coefficients in (6) are functions of x. Many of the equations that are of importance in mathematical physics are of this type. They are however,_ for the most part of the second order, and we shall accordingly confine our discussion to equations of this order.
Consider the equation goy" +piy'+p2y =f (x), (9) where the coefficients are either constants or functions of x. If we know a solution v of the homogeneous equation Po" +p1y'+p2y=0, (io) and if we introduce a new dependent variable z by virtue of the relation y = vz, we get Pon"
But v is a solution of (to). Hence
+
z' =f (x) • Now, although this last equation is of the second order, it con tains no term in z. If therefore we put t = z', we get
=f(x).
By the solution of this linear equation of the first order we get t. Then an integration gives z, and an algebraic operation gives y.
This method reduces the solution of linear equations of order greater than one to the solution of similar equations of order one less. The practical difficulty consists in finding a particular solution of the corresponding homogeneous equation. If the original equation is homogeneous we have a close analogy with the fact that the solution of an algebraic equation of degree n can be reduced to the solution of one of degree n — I when we know one root of the original equation.
(6) Equations of the first order that can be solved for y. These can be written in the form y = f(x, p), where p represents dy• dx dp By differentiation we get p =fx+ff • This is an equation of the first order and first degree in x and p. We can therefore solve it and get a relation between x and p. This relation, together with the original equation between y and p gives a parametric rep resentation of the integral curves, p playing the role of parameter.
If the
given equation can be solved for x in terms of y and p, a similar procedure can be followed.
It should be observed that an equation of the first order and the first degree does not have a singular solution.
This is a method for the approximate solution of differential equations. We refer the reader to the book by Runge cited in the bibliography at the end of this article for the details con cerning other methods, for this kind of solution.
The reader will have observed in the preceding discussion a complete lack of any general method of procedure for the solu tion of ordinary differential equations. A general method of approach to this problem has, however, been given by Sophus Lie (q.v.) in his theory of one-parameter groups. We refer the reader to the work of Lie mentioned in the bibliography.
Partial Differential Equations.—An equation involving certain independent and dependent variables and partial de rivatives of the dependent variables with respect to the independ ent variables is called a partial differential equation. In the dis cussion of partial differential equations we - hall confine our at tention to the case of two independent variables and one de pendent variable, although we shall have occasion to mention equations that contain more than two independent variables. The linear equation P =R, (II) ax ay where P, Q, and R are continuous functions of x, y and z and P and Q do not vanish simultaneously, is the simplest of these equations. Suppose that (I I) has a solution z=4)(x Now the direction cosines of the normal to the surface resented by this equation at the point (xo, Yo, zo) are proportional to the values of , and — at this point. Moreover, the ax values of P, Q, and R at (xo, Yo, zo) are proportional to the direction cosines of a line through this point, and the differential equation says that the normal to the surface at (xo, Yo, zo) is perpendicular to this line. That is, at every point of an integral surface the differential equation determines a unique line that is perpendicular to the normal to the surface at this point. These lines are tangent to the curves defined by the system of ordinary equations dx dy dz —=—=—• (12) P Q R -These curves are called the characteristic curves of the given dif ferential equation.
As a simple illustration we cite the equation x y =z. 03) ax ay The differential equations of the characteristic curves are dx dy dz = = (I4) where t is a parameter. The equations of the characteristic that passes through the point (xo, Yo, zo) are therefore x =xoet, y=yoet and z= zo et. If we let the point (xo, Yo, zo) vary along the curve y = x-F 2, z= x2— 1, we get a one-parameter family of char acteristic curves whose equations are If we eliminate xo and t from these equations, we get z = 4x2 2ky —x) This represents an integral surface through the given curve.
Since y—cix = o and z — c2x = o are solutions of (14), (Y,2±-= o) x x where ck(u , v) is an arbitrary function containing v, defines an integral surface of 03). But when we obtain an integral sur face in this way we do not know whether it passes through a given curve, or not.
We consider a curve on an integral surface whose tangent at a given point is an element of the cone corresponding to this point. The values of x, y, z, p, and q vary from point to point along this curve. It can be shown that they satisfy the system of ordinary equations dx dy dz dp dq — = = — — = — =dt (17) P Q pP-FqQ X-Fpz Y - F qZ where t is a parameter. These equations determine x, y, z, p and q as functions of t and an initial set of values of the dependent vari ables. The functions determining x, y and z define a set of curves that are known as the characteristic curves of the equation (15). If the equation is linear, these curves are the same as the charac teristic curves already defined for linear equations. The values of p and q determined by the remaining equations serve to orient a plane through the point (x,,, yo, zo). This plane is tangent to the integral surface on which the characteristic lies. The set of five values (x, y, z, p,q) are said to form a surface element and the set of surface elements along a characteristic curve are said to form a characteristic strip.
We have thus far looked upon the characteristics as deter mined by the integral surfaces. But the differential equations (17) of the characteristic strips can be written down directly without any reference to the integral surfaces. We then naturally enquire whether we can make use of the integrals of these equa tions in the determination of the integral surfaces. It turns out that this can be done, but we cannot go into the details here.
Equations of the Second Order.—From the point o'f view of mathematical physics the linear partial differential equations of the second order with two or three independent variables and one dependent variable are of great importance. We have space only for a brief reference to the more important types with brief statements of the physical problems that give rise to them.
(a) We mention first Laplace's equation in two and three dimensions, As an illustration of a physical problem that gives rise to an equation of this kind the following may be cited: Let K be a conductor of heat whose bounding surface is maintained at a constant temperature—that is, the temperature at any point of the surface is kept at a constant value U which varies con tinuously from point to point on the surface. After a time the temperature u at a given point within the conductor vvill no longer vary with the time, but will vary continuously from point to point in such a way that u will satisfy equation ( g).
The potential function due to the attraction of gravitating matter, or to an electric or magnetic field also satisfies (ig). For this reason a function of x, y and z that satisfies this equation is called a Newtonian potential function.
But in order to determine the temperature within the con ductor we must find not merely a solution of (ig), but a solution that is equal to U on the bounding surface. This gives rise to the important problem of finding a function of x, y and z that satisfies 09) within the region bounded by a given surface and has upon this surface prescribed values. This is known as Dirichlet' s problem.
The problem of the solution of Laplace's equation is closely connected with a problem in the calculus of variations (q. v.); namely, the problem of determining a function u of x, y and z that is continuous throughout a region V, together with its partial derivatives of the first two orders, and assumes prescribed values on the boundary, in such a way that it shall make the integral a minimum. A necessary condition that u shall have these properties is that it shall satisfy (iv).
Neumann's problem is similar to Dirichlet's with the difference that it requires the normal derivative of u to have assigned values on the bounding surface. Obviously, a solution of this problem can be determined only up to an additive constant. Moreover, if a is a solution, then all solutions that are subject to certain natural conditions are of the form u+c, where c is a constant. Both Laplace's and Neumann's problems can be solved by the methods of integral equations.
(b) D'Alembert's equation for the motion of a vibrating string is a2u_ 2 a2u at2 — a axe • This equation is of interest historically from the fact that it was the first partial differential equation to be studied by mathe maticians. It has as a solution u = f (x — at) +F(x+at), where f and F are arbitrary functions.
(c) The equation for the motion of a vibrating membrane is a2u_ 2 a2u a2u ate — + a .
y This involves three independent variables and one dependent variable.
(d) The equation of wave-propagation in three dimensions is a2u_ 2 a2u 02u • ate — a + d 2 + a y Here we have four independent variables.
(c) The equation for the flow of electricity in wires—the so called telegraph equation—is au a2u ate +b at — a2 axe .