SHORT COMMUNICATIONS

ON A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS WHOSE SOLUTION IS REDUCED TO THE INTEGRATION OF AN EQUATION OF CLAIRAUT TYPE

S. FEMPL

In his exhaustive paper [1] on the integration of the system of partial differential equations

\[ \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} = a(x,y)u+b(x,y)v+f(x,y), \]

\[ \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} = c(x,y)u+d(x,y)v+g(x,y) \]

Vekua showed that it can be written in the complex form

\[ \frac{\partial w}{\partial \bar z}=Aw+\overline{B}\,\overline{w}+\frac{F}{2}, \]

where

\[ z=x+yi,\quad \bar z=x-yi,\quad w=u+vi,\quad \bar w=u-vi,\quad A=(a+d+ic-ib)/4, \]

\[ B=(a-d+ic+ib)/4,\quad F=f+gi, \]

and gave its regular solutions.

The idea of reducing a system of equations to a single equation in complex form prompted me to try to apply this method to certain other systems and, if possible, to solve them.

In the present paper I give solutions of the system of equations

\[ u=xP+yQ+\operatorname{Re}\,[f(P+Qi)], \]

\[ v=xQ-yP+\operatorname{Im}\,[f(P+Qi)], \tag{1} \]

where

\[ P=\frac{1}{2}\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right),\quad Q=\frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right), \]

and \(f\) is a given analytic function.

Using the slightly modified [3] Pompeiu operator [2]

\[ B=\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}, \]

we therefore have

\[ \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} +i\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right) \equiv \frac{\partial w}{\partial x}+i\frac{\partial w}{\partial y} =B(w) \tag{2} \]

and obtain, after multiplying the second equation of (1) by \(i\) and adding it to the first,

\[ w=\frac{x}{2}B(w)-\frac{yi}{2}B(w)+f\left[\frac{1}{2}B(w)\right] \]

or

\[ w=\frac{\bar z}{2}B(w)+f\left[\frac{1}{2}B(w)\right]. \tag{3} \]

Thus, system (1) has been reduced to one equation in complex form, which resembles Clairaut’s differential equation.

If in the last equation we replace \(B(w)/2=p\), and then apply the operator \(B\) to it, then, owing to [4],

\[ B(w_1\cdot w_2)=w_1 B(w_2)+w_2 B(w_1) \tag{4} \]

and

\[ B[F(p)]=F'(p)B(p), \tag{5} \]

we obtain the following equation

\[ B(p)\,[\bar z+f'(p)]=0. \tag{6} \]

If \(B(p)=0\), then \(p\) is an analytic function, as is not hard to verify from (2). Thus, if we denote this arbitrary analytic function by
\(A(z)\equiv \alpha(x,y)+i\beta(x,y)\), then the solution of equation (3) can be written in the following form:

\[ w=\bar z A(z)+f[A(z)], \]

which gives solutions of the system of equations (1)

\[ u=\operatorname{Re}\{\bar z A(z)+f[A(z)]\}, \]

\[ v=\operatorname{Im}\{\bar z A(z)+f[A(z)]\}, \]

or, putting \(f[A(z)]=R+Ti\),

\[ u=\alpha x+\beta y+R,\quad v=\beta x-\alpha y+T, \tag{7} \]

where between the functions \(\alpha\) and \(\beta\) there must exist the relation

\[ \frac{\partial \alpha}{\partial x}=\frac{\partial \beta}{\partial y},\quad \frac{\partial \alpha}{\partial y}=-\frac{\partial \beta}{\partial x}, \]

since the function \(A(z)\) is analytic.

It is easy to show that the functions (7) do indeed represent solutions of the system of equations (1).

If in (6) \(\bar z=-f'(p)\), then it is not difficult to verify that this function, together with the function

\[ w=-pf'(p)+f(p), \]

which is obtained from (3), is also a solution of equation (3). Indeed, owing to

\[ \frac{dw}{dp}=-pf''(p),\quad -f''(p)\frac{dp}{dz}=1,\quad \frac{dw}{dz}=\frac{dw}{dp}\cdot\frac{dp}{dz}, \]

we obtain \(\dfrac{dw}{dz}=p\), and, taking into account that
\(2\dfrac{\partial}{\partial \bar z}=\dfrac{\partial}{\partial x}+i\dfrac{\partial}{\partial y}\) [1], we have \(2p=B(w)\).

Thus, the functions

\[ x-yi=-f'(p), \]

\[ u+vi=f(p)-pf'(p) \tag{8} \]

represent a singular solution of the given system, since they do not contain arbitrary functions \(\alpha\) and \(\beta\).

As an example, let us solve the system of equations

\[ u=xP+yQ+P^2-Q^2, \]

\[ v=xQ-yP+2PQ, \]

which gives

\[ u+vi=w=\frac{\bar z}{2}B(w)+\left[\frac{1}{2}B(w)\right]^2. \]

According to (7), here \(A(z)=\alpha+\beta i\), \(f[A(z)]=A^2(z)\). Thus \(R=\alpha^2-\beta^2\), \(T=2\alpha\beta\), and we obtain the solutions

\[ u=\alpha x+\beta y+\alpha^2-\beta^2,\quad v=\beta x-\alpha y+2\alpha\beta, \]

where the arbitrary functions \(\alpha(x,y)\) and \(\beta(x,y)\) are connected by the Cauchy–Riemann conditions.

A special solution of this system is obtained according to (8) and, bearing in mind that \(f(t) \equiv t^2\), from the equations

\[ x - yi = -2p, \]

\[ u + vi = p^2 - 2p^2 = -p^2, \]

i.e.

\[ u = -\frac{1}{4}(x^2 - y^2), \qquad v = -\frac{1}{2}xy. \]

References

1. Vekua I. N. Matem. sb., 31 (73). Moscow, 1952, pp. 217–314.
2. Pompeiu D. Rendiconti del Circolo Matematico di Palermo, 33, 1912, pp. 108–113 and 35, 1913, pp. 277–281.
3. Bilimovich A. Differential elements of the geometric theory of nonanalytic functions, “Glas” of the Serbian Academy of Sciences, CCXLII, 19. Belgrade, 1960, 1–82.
4. Fempl S. Matematički vesnik, 1 (16). Belgrade, 1964, pp. 29–38.

Received by the editors
January 14, 1965

Belgrade,
Yugoslavia