Simultaneous Systems in Three Variables

Before the general theory of the integration of simultaneous systems of differential equations is attacked, it will be convenient to dispose of a simple case in which the equations are integrable by the methods which were detailed in the earlier sections of the chapter.

Consider the system

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta },$$

$\xi ,\eta$ and $\zeta$ are, in general, functions of $x,y$ and $z$. A very special, but important case is that in which $\xi$ and $\eta$ are independent of $z$. In this case the equation

$$\frac{dx}{\xi }=\frac{dy}{\eta }$$

involves only $x$ and $y$; it will be supposed that this equation can be integrated and that its solution is

$$\mathrm{\Phi }(x,y,\alpha )=0,$$

where $\alpha$ is the constant of integration. Let this equation be solved for $y$, thus

$$y=\psi (x,\alpha ),$$

and let ${\xi }_1$ and ${\zeta }_1$ be what $\xi$ and $\zeta$ become when $y$ is replaced therein by $\psi (x,\alpha )$. Then the equation

$$\frac{dx}{{\xi }_1}=\frac{dz}{{\zeta }_1}$$

does not involve $y$. Its solution will be of the form

$$\mathrm{\Theta }(x,z,\alpha ,\beta )=0,$$

where $\beta$ is the constant of integration. Now let $\alpha$ be eliminated between the two solutions

$$\mathrm{\Phi }(x,y,\alpha )=0,\mathrm{\Theta }(x,z,\alpha ,\beta )=0;$$

the solutions then take the form

$$\mathrm{\Phi }(x,y,\alpha )=0,\mathrm{\Psi }(x,y,z,\beta )=0.$$

2.7.1 Integration of a Simultaneous Linear System with Constant Co-efficients

The system

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta }$$

where

\begin{align*} \xi &= a_1x+b_1y+c_1z+d_1, \\ \eta &= a_2x+b_2y+c_2z+d_2, \\ \zeta &= a_3x+b_3y+c_3z+d_3, \end{align*}

is not of the form dealt with in the preceding section. It can, however, be dealt with in a similar manner after a linear transformation of the variables has been made. To simplify the working a new variable $t$ is introduced such that

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta }=\frac{dt}{t},$$

then, whatever constants $l,m,n$ may be,

$$\frac{dt}{t}=\frac{ldx+mdy+ndz}{l\xi +m\eta +n\zeta }.$$

Let $l,m,n$ be so chosen that

\begin{align*} la_1+ma_2+na_3 &= l\rho, \\ lb_1+mb_2+nb_3 &= m\rho, \\ lc_1+mc_2+nc_3 &= n\rho, \end{align*}

then

$$\frac{dt}{t}=\frac{d(lx+my+nz)}{\rho (lx+my+nz+r)},$$

where $r\rho =l{d}_1+m{d}_2+n{d}_3$. This choice of $l,m,n$ is possible if $\rho$ is a root of the equation

Let the roots of this equation, supposed distinct be $\frac{1}{{\lambda }_1},\frac{1}{{\lambda }_2},\frac{1}{{\lambda }_3}$, and let the corresponding values of $l,m,n,r$ be

$$l_i,m_i,n_i,r_i(i=1,2,3),$$

then

$$\frac{dt}{t}=\frac{{\lambda }_{i}d(l_{i}x+m_{i}y+n_{i}z)}{l_{i}x+m_{i}y+n_{i}z+r_i},$$

whence

$$t=C_i(l_{i}x+m_{i}y+n_{i}z+r_i{)}^{{\lambda }_i}.$$

The solution of the system is therefore

$$C_1(l_{1}x+m_{1}y+n_{1}z+r_1{)}^{{\lambda }_1}=C_2(l_{2}x+m_{2}y+n_{2}z+r_2{)}^{{\lambda }_2}=C_3(l_{3}x+m_{3}y+n_{3}z+r_3{)}^{{\lambda }_3}$$

and contains three constants of integration, $C_1,C_2,C_3$, of which two are arbitrary.

2.7.2 The Equivalent Partial Differential Equation

Let $x$ and $y$ be regarded as independent variables, and $z$ as a dependent variable. Let $p$ and $q$ be the partial derivatives of $z$ with respect to $x$ and $y$ respectively, then

$$\xi p+\eta q=\zeta$$

is a linear partial differential equation of the first order and is known as the Lagrange linear equation. If

$$z=f(x,y)$$

is a solution of the equation, then

$$\xi \frac{\partial f}{\partial x}+\eta \frac{\partial f}{\partial y}=\zeta ,$$

for all values of $x,y$. This solution represents a surface, known as an integral-surface of the partial differential equation. Since the direction cosines of the normal to a surface $z=f(x,y)$ are proportional to

$$\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},-1,$$

the differential equation expresses a distinguishing property of the tangent plane to the integral-surface.

Now consider the system of simultaneous ordinary differential equations

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta },$$

and let its solutions be solved for the constants of integration, thus

$$u(x,y,z)=\alpha ,v(x,y,z)=\beta .$$

These solutions represent a two-parameter family of curves in space, which are known as the characteristics of the system. If $\xi ,\eta ,\zeta$ exist and are one-valued at a point $(x_0,y_0,z_0)$, and at least one of them is not zero at $(x_0,y_0,z_0)$, one and only one characteristic passes through that point.

It will now be shown that the characteristics of the simultaneous differential system bear an intimate relationship to the integral-surface of the partial differential equation. In the first place it will be proved that, if an integral-surface passes through $(x_0,y_0,z_0)$, it contains the characteristic through that point. Let the integral surface through $(x_0,y_0,z_0)$ be

$$z=f(x,y)$$

and, supposing that $\xi$ does not vanish at $(x_0,y_0,z_0)$, consider the differential equation

$$\frac{dy}{dx}=\frac{\eta }{\xi },$$

in which $z$ has been replaced by $f(x,y)$. The equation defines $y$ as a function of $x$ and is therefore the differential equation of a family of cylinders whose generators are parallel to the axis of $z$. The cylinder through $(x_0,y_0,0)$ intersects the integral-surface in a curve through $(x_0,y_0,z_0)$. Along this curve

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{pdx+qdy}{p\xi +q\eta }=\frac{dz}{\zeta }.$$

The curve so defined is therefore a characteristic, and the theorem is proved. An immediate consequence of this theorem is the fact that every integral surface is a locus of characteristics. In particular if any non-characteristic curve in space is drawn, the characteristics which pass through the points of this curve build up an integral-surface.

In the second place, the converse of this theorem will be shown to be true, namely, that in general every surface which arises as a locus of characteristic curves is an integral-surface of the partial differential equation.¹ The tangent line to the characteristic at any arbitrary point $(x_0,y_0,z_0)$ is

$$\frac{x-x_0}{{\xi }_0}=\frac{y-y_0}{{\eta }_0}=\frac{z-z_0}{{\zeta }_0},$$

where ${\xi }_0,{\eta }_0,{\zeta }_0$ are the values of $\xi ,\eta ,\zeta$ at $(x_0,y_0,z_0)$. The equation of the tangent plane at $(x_0,y_0,z_0)$ to the surface which envelopes the characteristics will be

$$(x-x_0)p_0+(y-y_0)q_0=z-z_0,$$

where $p_0$ and $q_0$ are respectively the values of $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$ on the surface at $(x_0,y_0,z_0)$. Since the characteristic lies in the surface, the tangent line lies in the tangent plane, and therefore

$${\xi }_0{p}_0+{\eta }_0{q}_0={\zeta }_0.$$

But $(x_0,y_0,z_0)$ is any point on the surface; the latter is therefore an integral-surface of the partial differential equation

$$\xi p+\eta q=\zeta .$$

2.7.3 Formation of the Integral-Surface

The aggregate of characteristics form a two-parameter family or congruence of curves. Just as a plane curve is formed by selecting, according to a definite law, a one-fold infinity of the two-fold infinity of points in a plane, so an integral-surface is formed by selecting a one-fold infinity of curves of the congruence. Let

$$u(x,y,z)=\alpha ,v(x,y,z)=\beta$$

be the aggregate of characteristics from which a one-fold infinity is chosen by setting up a relationship between $\alpha$ and $\beta$, say

$$\mathrm{\Omega }(\alpha ,\beta )=0.$$

The equation to the integral-surface is therefore

$$\mathrm{\Omega }(u,v)=0,$$

and this equation, in which the function $\mathrm{\Omega }$ is arbitrary, is the general solution of the partial differential equation.

In the theory of ordinary differential equations of the first order, it is often required to find that integral-curve which passes through a given point of the plane. The corresponding problem in the case of partial differential equation is to find that integral-surface which passes through a given (non-characteristic) base-curve in space. This problem, in its general form, is known as Cauchy's problem.

Let

$$\varphi (x,y,z)=0,\psi (x,y,z)=0$$

represent the base-curve, and let

$$u(x,y,z)=\alpha ,v(x,y,z)=\beta$$

be the characteristics. If, between these four equations, $x,y,z$ are eliminated, there remains a relation between $\alpha$ and $\beta$ which expresses the condition that the characteristics and the base-curve have points in common. Let this relation be

$$\mathrm{\Phi }(\alpha ,\beta )=0,$$

then

$$\mathrm{\Phi }(u,v)=0$$

is the required integral-surface.

2.7.4 The Homogeneous Linear Partial Differential Equation

When $\zeta$ is identically zero, the equation has the so-called homogeneous form

$$\xi p+\eta q=0.$$

The equations of the characteristics then become

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{0}.$$

The last equation gives at once

$$z=\alpha ,$$

and therefore the characteristics are plane curves whose planes are perpendicular to the $z$-axis.

The most important case is that in which $\xi$ and $\eta$ are independent of $z$; the equation of the characteristics is then

$$z=\alpha ,u(x,y)=\beta ,$$

and the equation of the integral-surface may be written in the form

$$z=f(u).$$

Now consider the equation

$$\xi \frac{\partial f}{\partial x}+\eta \frac{\partial f}{\partial y}+\zeta \frac{\partial f}{\partial z}=0,$$

where $\xi ,\eta ,\zeta$ are functions of $x,y,z$ and do not involve $f$. If

$$f(x,y,z)=c,$$

where $c$ is a constant, is a solution of the partial differential equation, then

$$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz=0,$$

and therefore $f(x,y,z)=c$ is a solution of the simultaneous system

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta }.$$

The converse is also true, for if

$$u(x,y,z)=\alpha$$

is any solution of the simultaneous system, then

$$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz=0,$$

and therefore

$$\xi \frac{\partial u}{\partial x}+\eta \frac{\partial u}{\partial y}+\zeta \frac{\partial u}{\partial z}=0.$$

Let

$$v(x,y,z)=\beta$$

be a second, and distinct, solution of the simultaneous system; it will also be a solution of the partial differential equation, so that

$$\xi \frac{\partial v}{\partial x}+\eta \frac{\partial v}{\partial y}+\zeta \frac{\partial v}{\partial z}=0.$$

If any other solution

$$w(x,y,z)=\gamma$$

exists, then

$$\xi \frac{\partial w}{\partial x}+\eta \frac{\partial w}{\partial y}+\zeta \frac{\partial w}{\partial z}=0,$$

and, eliminating $\xi ,\eta ,\zeta$,

identically. Consequently $w$ is a function of $u$ and $v$, and therefore the partial differential equation admits of two and only two distinct solutions.

From the three equations

$$u(x,y,z)=\alpha ,v(x,y,z)=\beta ,w(x,y,z)=\gamma ,$$

two of the variables, say $x$ and $y$, may be eliminated, and the eliminant can be expressed in the form

$$w=\varphi (u,v,z).$$

Now

$$0=\frac{\partial (u,v,w)}{\partial (x,y,z)}=\frac{\partial (u,v,\varphi )}{\partial (u,v,z)}\frac{\partial (u,v,z)}{\partial (x,y,z)}.$$

The first determinant on the right is simply $\frac{\partial \varphi }{\partial z}$, the second is $\frac{\partial (u,v)}{\partial (x,y)}$. The second of these is not zero, since $u$ and $v$ are supposed to be independent. Consequently

$$\frac{\partial \varphi }{\partial z}=0,$$

that is to say, $\varphi$ is explicitly independent of $z$, or in other words $w$ is a function of $u$ and $v$ alone.

The general solution of the partial differential equation

$$\xi \frac{\partial f}{\partial x}+\eta \frac{\partial f}{\partial y}+\zeta \frac{\partial f}{\partial z}=0$$

is therefore

$$\mathrm{\Omega }(u,v)=\text{const.},$$

where $\mathrm{\Omega }$ is an arbitrary function of its arguments, and

$$u=\alpha ,v=\beta$$

are any two independent solutions of the subsidiary system

$$\frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dz}{\zeta }.$$

The extension to the case of $n$ variables is obvious. An exceptional case occurs when $\xi ,\eta ,\zeta$ have a common factor; the result of equating this factor to zero provides a special solution of the partial differential equation which may or may not be included in the general solution.

Footnotes

1. The exceptional case arises when the surface has a tangent plane parallel to the $z$-axis, for then $p$ and $q$ become infinite and the proof fails. ↩