Total Differential Equations

An algebraic equation in three variables, of the form

$$\varphi (x,y,z)=c,$$

where $c$ is a constant, leads to the total differential equation

$$\frac{\partial \varphi }{\partial x}dx+\frac{\partial \varphi }{\partial y}dy+\frac{\partial \varphi }{\partial z}dz=0.$$

If $\frac{\partial \varphi }{\partial x},\frac{\partial \varphi }{\partial y},\frac{\partial \varphi }{\partial z}$ have a common factor $\mu$, and if

$$\frac{\partial \varphi }{\partial x}=\mu P,\frac{\partial \varphi }{\partial y}=\mu Q,\frac{\partial \varphi }{\partial z}=\mu R,$$

the total differential equation may be written in the form

$$Pdx+Qdy+Rdz=0.$$

On the other hand, if $P,Q$, and $R$ are arbitrarily-assigned functions of $x,y,z$, the total differential equation does not necessarily correspond to a primitive of the form

$$\varphi (x,y,z)=c.$$

For if such a primitive exists, $P,Q,R$ are respectively proportional to the three partial differential coefficients of a function $\varphi (x,y,z)$, which is not in general true. The problem therefore arises, to find a necessary and sufficient condition that a given total differential equation should be integrable, that is to say, derived from a primitive of the form considered.

It is first of all necessary that functions $\varphi (x,y,z)$ and $\mu (x,y,z)$ exist such that the conditions

$$\frac{\partial \varphi }{\partial x}=\mu P,\frac{\partial \varphi }{\partial y}=\mu Q,\frac{\partial \varphi }{\partial z}=\mu R$$

are satisfied. Then¹

$$\frac{\partial }{\partial y}(\mu P)=\frac{{\partial }^2\varphi }{\partial y\partial x}=\frac{{\partial }^2\varphi }{\partial x\partial y}=\frac{\partial }{\partial x}(\mu Q).$$

that is

$$\mu (\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})=Q\frac{\partial \mu }{\partial x}-P\frac{\partial \mu }{\partial y},$$

and similarly

$$\mu (\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y})=R\frac{\partial \mu }{\partial y}-Q\frac{\partial \mu }{\partial z},$$$$\mu (\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z})=P\frac{\partial \mu }{\partial z}-R\frac{\partial \mu }{\partial x}.$$

The unknown $\mu$ is eliminated from these three equations by multiplying respectively by $R,P,Q$ and adding. The resulting equation

$$P(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y})+Q(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z})+R(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})=0$$

is a necessary condition for integrability.²

It is obvious from the above demonstration, and may easily be verified independently, that if $\lambda$ is a function of $x,y,z$ and

$$P_1=\lambda P,Q_1=\lambda Q,R_1=\lambda R.$$

the condition for integrability is satisfied by $P_1,Q_1,R_1$.

It will now be proved that the condition of integrability is a sufficient condition, that is to say, when it is satisfied, there exists a solution involving an arbitrary constant. The proof incidentally furnishes a method of obtaining the solution when the condition for integrability is satisfied.

Let one of the variables be, for the moment, regarded as a constant. If the variable chosen is $z$, the equation reduces to

$$Pdx+Qdy=0,$$

where $P$ and $Q$ are to be regarded as functions of $x$ and $y$ into which $z$ enters as a parameter. This equation has a solution

$$u(x,y,z)=\text{const.}$$

where, if $\lambda (x,y,z)$ is the integrating factor,

$$\frac{\partial u}{\partial x}=\lambda P=P_1,\frac{\partial u}{\partial y}=\lambda Q=Q_1,$$

but, of course, it does not follow that

$$\frac{\partial u}{\partial z}=\lambda R=R_1.$$

Let

$$R_1=\lambda R=\frac{\partial u}{\partial z}+S,$$

then since, by hypothesis,

$$P_1(\frac{\partial Q_1}{\partial z}-\frac{\partial R_1}{\partial y})+Q_1(\frac{\partial R_1}{\partial x}-\frac{\partial P_1}{\partial z})+R_1(\frac{\partial P_1}{\partial y}-\frac{\partial Q_1}{\partial x})=0,$$

it follows that

$$\frac{\partial S}{\partial y}\frac{\partial u}{\partial x}-\frac{\partial S}{\partial x}\frac{\partial u}{\partial y}=0.$$

This relation is not satisfied in virtue of

$$u(x,y,z)=\text{const.},$$

it is therefore an identity. Consequently $S$ and $u$, regarded as functions of $x$ and $y$ are functionally dependent upon one another. The functional relationship between them, however, involves also the third variable $z$, and thus $S$ is expressible in terms of $u$ and $z$ alone.

Now

$$\lambda (Pdx+Qdy+Rdz)=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+(\frac{\partial u}{\partial z}+S)dz=du+Sdz.$$

The original equation is therefore equivalent to

$$du+Sdz=0;$$

let $\mu (u,z)$ be an integrating factor, then

$$\lambda \mu (Pdx+Qdy+Rdz)=\mu (du+Sdz)$$

is an exact differential $d\psi$. The primitive is

$$\psi (u,z)=c;$$

and if $u$ is replaced by its expression in $x,y,z$ the primitive takes the form

$$\varphi (x,y,z)=c.$$

Similarly it may be proved that a necessary and sufficient condition that the equation in $n$ variables

$$X_{1}d{x}_1+X_{2}d{x}_2+\cdots +X_{n}d{x}_n=0$$

should have a primitive of the form

$$\varphi (x_1,x_2,\dots ,x_n)=c$$

is that the set of equations

$$X_{\nu }(\frac{\partial X_{\mu }}{\partial x_{\lambda }}-\frac{\partial X_{\lambda }}{\partial x_{\mu }})+X_{\lambda }(\frac{\partial X_{\nu }}{\partial x_{\mu }}-\frac{\partial X_{\mu }}{\partial x_{\nu }})+X_{\mu }(\frac{\partial X_{\lambda }}{\partial x_{\nu }}-\frac{\partial X_{\nu }}{\partial x_{\lambda }})=0$$

$(\lambda ,\mu ,\nu =1,2,\dots ,n)$, are satisfied simultaneously and identically. The total number of such equations is $\frac{1}{6}n(n-1)(n-2)$; of these $\frac{1}{2}(n-1)(n-2)$ are independent.

The main lines upon which the integration proceeds is illustrated by the following example:

2.8.1 Geometrical Interpretation

When $R$ is not zero, the total differential equation may be written

$$dz=-\frac{P}{R}dx-\frac{Q}{R}dy$$

or

$$dz=Udx+Vdy.$$

Since

$$dz=pdx+qdy,$$

the total differential equation is equivalent to the two simultaneous partial differential equations

$$p=U(x,y,z),q=V(x,y,z).$$

The equation of the tangent plane at $(x_0,y_0,z_0)$ to the integral-surface which passes through $(x_0,y_0,z_0)$ is therefore,

$$z-z_0=U_0(x-x_0)+V_0(y-y_0),$$

where $U_0$ and $V_0$ are respectively the values of $U$ and $V$ at $(x_0,y_0,z_0)$.

The problem of integration is therefore equivalent to finding a surface such that the direction cosines of its normal at every point $(x,y,z)$ are proportional to

$$U(x,y,z),V(x,y,z),-1.$$

This problem is, in general, insoluble; in order that it may be soluble the condition for integrability, which reduces to

$$\frac{\partial U}{\partial y}+V\frac{\partial U}{\partial z}=\frac{\partial V}{\partial x}+U\frac{\partial V}{\partial z},$$

must be satisfied.

The general solution of each of the partial differential equations

$$\frac{\partial z}{\partial x}=U,\frac{\partial z}{\partial y}=V$$

represents a family of surfaces, such that through every curve in space there passes, in general, one and only one surface of each family.³ Their common solution represents a family of space-curves

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

depending upon the two parameters $\alpha$ and $\beta$, and such that through each point in space then passes one and only one integral-curve.

An integral-surface of the total differential equation cuts every curve of this family orthogonally, that is the tangent plane at any point $P$ of an integral-surface must contain the normals at $P$ of the two surfaces $u=\alpha ,v=\beta$ which pass through $P$. Hence

$$p\frac{\partial u}{\partial x}+q\frac{\partial u}{\partial y}-\frac{\partial u}{\partial z}=0,$$$$p\frac{\partial v}{\partial x}+q\frac{\partial v}{\partial y}-\frac{\partial v}{\partial z}=0.$$

These two equations determine

$$p=U(x,y,z),q=V(x,y,z).$$

These are consistent if, and only if

$$\frac{\partial p}{\partial y}=\frac{\partial q}{\partial x},$$

that is, if the condition for integrability

$$\frac{\partial U}{\partial y}+V\frac{\partial U}{\partial z}=\frac{\partial V}{\partial x}+U\frac{\partial V}{\partial z}$$

is satisfied.

2.8.2 Mayer's Method of Integration

The method of integration developed in § 2.8 depends upon the integration of two successive differential equations in two variables. In Mayer's method⁴ only one integration is necessary. Let $(x_0,y_0)$ be any chosen pair of values of $(x,y)$ and let $z_0$ be an arbitrary value of $z$ such that the four differential coefficients

$$\frac{\partial U}{\partial y},\frac{\partial U}{\partial z},\frac{\partial V}{\partial x},\frac{\partial V}{\partial z}$$

exist and are continuous in the neighbourhood of $(x_0,y_0,z_0)$. Then if the equation is integrable, its solution will be completely determined by the initial value $z_0$. The value of $z$ at $(x,y)$ can therefore be obtained by following the variation of $z$ from its initial value $z_0$ as a point $P$ moves in a straight line in the $(x,y)$-plane from $(x_0,y_0)$ to $(x,y)$.

There is no loss in generality in supposing that the point $(x_0,y_0)$ is the origin, and this will be assumed. On the straight line joining the origin to $(x,y)$,

$$y=\kappa x,dy=\kappa dx,$$

where $\kappa$ is constant. The equation therefore becomes

$$dz=(U_1+\kappa V_1)dx,$$

where $U_1$ and $V_1$ are what $U$ and $V$ become when $y$ is replaced by $\kappa x$. This equation, in the two variables $x$ and $z$, has a solution of the form

$$\varphi (x,z,\kappa )=\text{const.}$$

or, since $z=z_0$ when $x=0$,

$$\varphi (x,z,\kappa )=\varphi (0,z_0,\kappa ).$$

On replacing $\kappa$ by $y/x$, the solution

$$\varphi (x,z,y/x)=\varphi (0,z_0,y/x)$$

is obtained in a form which indicates its dependence upon the arbitrary constant $z_0$.

2.8.3 Pfaff's Problem

When the condition for integrability is not satisfied, the total differential equation is not derivable from a single primitive. On this account such an equation was at one time regarded as meaningless.⁵ Further consideration, however, brought to light the fact that the total differential equation is equivalent to a pair of algebraic equations⁶ known as its integral equivalents. In general, when the equations for integrability are not all satisfied, a total differential equation in $2n$ or $2n-1$ variables is equivalent to a system of not more than $n$ algebraic equations.⁷ The problem of determining the integral equivalents of any given total differential equation is known as Pfaff's Problem. A sketch of the method of procedure, in the case of three variables, will now be given.⁸

The first step consists in showing that the differential expression

$$Pdx+Qdy+Rdz$$

can be reduced to the form

$$du+vdw,$$

where $u,v,w$ are functions of $x,y,z$. The two forms are identical if

\begin{cases} P = \frac{\partial u}{\partial x} + v\frac{\partial w}{\partial x}, \\ Q = \frac{\partial u}{\partial y} + v\frac{\partial w}{\partial y}, \\ R = \frac{\partial u}{\partial z} + v\frac{\partial w}{\partial z}. \end{cases} \tag{A}

Let

P' = \frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y}, \quad Q' = \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}, \quad R' = \frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x},

then

\begin{align*} P' &= \frac{\partial v}{\partial z}\frac{\partial w}{\partial y} - \frac{\partial v}{\partial y}\frac{\partial w}{\partial z}, \\ Q' &= \frac{\partial v}{\partial x}\frac{\partial w}{\partial z} - \frac{\partial v}{\partial z}\frac{\partial w}{\partial x}, \\ R' &= \frac{\partial v}{\partial y}\frac{\partial w}{\partial x} - \frac{\partial v}{\partial x}\frac{\partial w}{\partial y}. \end{align*}

It follows that

P'\frac{\partial v}{\partial x} + Q'\frac{\partial v}{\partial y} + R'\frac{\partial v}{\partial z} = 0,P'\frac{\partial w}{\partial x} + Q'\frac{\partial w}{\partial y} + R'\frac{\partial w}{\partial z} = 0.

Thus $v$ and $w$ are solutions of one and the same linear partial differential equation; the equivalent simultaneous system is

\frac{dx}{P'} = \frac{dy}{Q'} = \frac{dz}{R'}.

Let

$$\alpha (x,y,z)=\text{const.},\beta (x,y,z)=\text{const.}$$

be two independent solutions of the simultaneous system, then $v$ and $w$ are functions of $\alpha$ and $\beta$.

Now return to the variable $u$; since

P'\left(P-\frac{\partial u}{\partial x}\right) + Q'\left(Q-\frac{\partial u}{\partial y}\right) + R'\left(R-\frac{\partial u}{\partial z}\right) = v\left(P'\frac{\partial w}{\partial x} + Q'\frac{\partial w}{\partial y} + R'\frac{\partial w}{\partial z}\right) = 0,

it follows that

P'\frac{\partial u}{\partial x} + Q'\frac{\partial u}{\partial y} + R'\frac{\partial u}{\partial z} = PP' + QQ' + RR'.

But the condition

PP'+QQ'+RR'=0

is the condition for integrability; since it is supposed not to be satisfied, $u$ does not satisfy the same partial differential equation as $v$ and $w$.

Now $w$ may be any function of $\alpha$ and $\beta$; for simplicity let

$$w=\alpha .$$

Then if the relation

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

where $a$ is a constant, is set up between the variables $x,y,z$, the differential form $Pdx+Qdy+Rdz$ reduces to $du$, and therefore becomes a perfect differential. Thus the relation $\alpha (x,y,z)=a$ is used to express any variable, say $z$, and its differential $dz$ in terms of the other two variables and their differentials, and when these expressions are substituted for $z$ and $dz$ in $Pdx+Qdy+Rdz$, the latter becomes a total differential $d\varphi (x,y,a)$. When $a$ is replaced by $\alpha (x,y,z)$ this differential becomes $du$. Thus $u$ is obtained, and since $u$ and $w$ are known, $v$ may be deduced algebraically from any one of the equations (A). The total differential equation

$$Pdx+Qdy+Rdz=0$$

is thus reduced to the canonical form

$$du+vdw=0.$$

The canonical equation may be satisfied in various ways, as follows:
(i) $u=\text{const.},w=\text{const.}$
(ii) $u=\text{const.},v=0.$
More generally, if $\psi (u,w)$ is any arbitrary function of $u$ and $w$, an integral equivalent is
(iii) $\psi (u,w)=0,v\frac{\partial \psi }{\partial u}-\frac{\partial \psi }{\partial w}=0$;
(iii) includes (ii) but not (i). In each case, the integral equivalent consists of a pair of algebraic equations.

2.8.4 Reduction of an Integrable Equation to Canonical Form

The foregoing reduction to canonical form may equally well be performed in the case of an integrable equation, but since, in this case,

PP'+QQ'+RR'=0,

identically, $u$ satisfies the same partial differential equation as $v$ and $w$ and therefore $u,v$ and $w$ are functions of $\alpha$ and $\beta$.

It follows that

$$du+vdw=Ad\alpha +Bd\beta ,$$

where $A$ and $B$ are functions of $\alpha$ and $\beta$ alone. When $\alpha$ and $\beta$ have been determined, $A$ and $B$ are derivable algebraically from any two of the three consistent equations,

$$P=A\frac{\partial \alpha }{\partial x}+B\frac{\partial \beta }{\partial x},Q=A\frac{\partial \alpha }{\partial y}+B\frac{\partial \beta }{\partial y},R=A\frac{\partial \alpha }{\partial z}+B\frac{\partial \beta }{\partial z}.$$

Thus the total differential equation is transformed into an ordinary equation in the two variables $\alpha$ and $\beta$.

This leads to a practical method of solving an integrable equation, as is shown by the following example (cf. § 2.8):

Footnotes

1. It is, of course, assumed that the change of order of differentiation is valid. ↩

2. Euler, Inst. Calc. Int. 3 (1770), p. 1. ↩

3. This depends upon the fact that a partial differential equation possesses, in general, a unique solution satisfying assigned initial conditions. The truth of the underlying existence-theorem is assumed. ↩

4. Math. Ann., 5 (1872), p. 448. ↩

5. Euler, Inst. Calc. Int., 3 (1770), p. 5. ↩

6. Monge, Mém. Acad. Sc. Paris (1784), p. 535. ↩

7. Pfaff, Abh. Akad. Wiss. Berlin (1814), p. 76. ↩

8. An extended treatment in the general case is given in Forsyth, Theory of Differential Equations, Part I., and in Goursat, Leçons sur le Problème de Pfaff. ↩