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MATHEMATICS
S. V. BAKHVALOV
ON THE NOMOGRAPHING OF SOLUTIONS OF A SECOND-ORDER DIFFERENTIAL EQUATION
(Presented by Academician P. S. Aleksandrov, 3 III 1962)
The second-order differential equation
\[ \frac{d^2 v}{d u^2}=F\left(u, v, \frac{d v}{d u}\right) \tag{1} \]
will be called nomographable under the boundary conditions
\[ u=u_1,\quad v=v_1 \quad \text{and} \quad u=u_2,\quad v=v_2, \tag{2} \]
if there exists a Massau determinant
\[ \left| \begin{array}{ccc} A_1(u) & A_2(u) & A_3(u)\\ B_1(v) & B_2(v) & B_3(v)\\ c_1 & c_2 & 1 \end{array} \right| \not\equiv 0, \tag{3} \]
satisfying the condition
\[ \left| \begin{array}{ccc} A_1(u) & A_2(u) & A_3(u)\\ B_1(\bar v) & B_2(\bar v) & B_3(\bar v)\\ c_1 & c_2 & 1 \end{array} \right| \equiv 0, \tag{4} \]
where \(\bar v=v(u,c_1,c_2)\) is the general solution of equation (1), and \(c_1,c_2\) are arbitrary constants. In this case the point \(M\) with mark \(u\) of the scale \(u\), the point \(N\) with mark \(v\) of the scale \(v\), and the point \(C\) with coordinates \((c_1;c_2)\) lie on one straight line. The point \(C(c_1;c_2)\) is determined, by means of the boundary conditions (2), as the point of intersection of the straight lines \(M_1N_1\) and \(M_2N_2\), passing respectively through the points \(M_1(u_1), N_1(v_1)\) and \(M_2(u_2), N_2(v_2)\) (see Fig. 1).
Fig. 1
Theorem. A necessary and sufficient condition for the nomographability of equation (1) is that its general solution be a solution of the differential equation of geodesic lines on a sphere under the conditions
\[ \left\{\begin{array}{c} 11\\ 2 \end{array}\right\}=0,\qquad \left\{\begin{array}{c} 22\\ 1 \end{array}\right\}=0. \tag{5} \]
- Suppose that equation (1) is nomographable. Then it follows from (4) that its general solution has the form
\[ c_1 P_{23}(u,v)+c_2 P_{31}(u,v)+P_{12}(u,v)=0, \tag{6} \]
where
\[ P_{ik}= \left| \begin{array}{cc} A_i & A_k\\ B_i & B_k \end{array} \right|. \tag{7} \]
It is known that the vector \(\mathbf P=\{P_{23};P_{31};P_{12}\}\) satisfies conditions (2)
\[ (\mathbf P\mathbf P'_u\mathbf P''_{uu}) \equiv 0, \tag{8} \]
\[ (\mathbf P\mathbf P'_v\mathbf P''_{vv}) = 0, \tag{9} \]
\[ (\mathbf P\mathbf P'_u\mathbf P''_v) \ne 0. \tag{10} \]
Conditions (8), (9), (10) are necessary and sufficient for the left-hand side of equation (6) to be representable in the form of the Massau determinant (3).
From (6) we obtain
\[ \begin{aligned} c_1q_{23}+c_2q_{31}+q_{12}&=0,\\ c_1r_{23}+c_2r_{31}+r_{12}&=0, \end{aligned} \tag{11} \]
where
\[ q_{ik}=\frac{\partial P_{ik}}{\partial u}+\frac{\partial P_{ik}}{\partial v}\frac{dv}{du}, \]
\[ r_{ik}=\frac{\partial^2 P_{ik}}{\partial u^2} +2\frac{\partial^2 P_{ik}}{\partial u\,\partial v}\frac{dv}{du} +\frac{\partial^2 P_{ik}}{\partial v^2}\left(\frac{dv}{du}\right)^2 +\frac{\partial P_{ik}}{\partial v}\frac{d^2v}{du^2}. \tag{12} \]
Eliminating \(c_1,c_2\) from (6), (11), we obtain the desired differential equation
\[ \frac{d^2v}{du^2}(\mathbf P\mathbf P'_u\mathbf P'_v) +\left(\frac{dv}{du}\right)^3(\mathbf P\mathbf P'_v\mathbf P''_{vv}) +\left(\frac{dv}{du}\right)^2\left[(\mathbf P\mathbf P'_u\mathbf P''_{vv}) +2(\mathbf P\mathbf P'_v\mathbf P''_{uv})\right]+ \]
\[ +\left(\frac{dv}{du}\right)\left[(\mathbf P\mathbf P'_v\mathbf P''_{uu}) +2(\mathbf P\mathbf P'_u\mathbf P''_{uv})\right] +(\mathbf P\mathbf P'_u\mathbf P''_{uu})=0. \tag{13} \]
Consider the functions \(x=\lambda(u,v)P_{23},\ y=\lambda(u,v)P_{31},\ z=\lambda(u,v)P_{12}\). Choose \(\lambda(u,v)\) so that \(x^2+y^2+z^2=1\). Then \(x,y,z\) are the coordinates of a point of the sphere whose radius is equal to unity. If, in equations (8), (9), (10), (13), \(P_{23},P_{31},P_{12}\) are replaced respectively by \(x/\lambda,\ y/\lambda,\ z/\lambda\), then these equations will have the form:
\[ (\vec\rho\,\vec\rho'_u\vec\rho''_{uu}) \equiv 0, \tag{8'} \]
\[ (\vec\rho\,\vec\rho'_v\vec\rho''_{vv}) \equiv 0, \tag{9'} \]
\[ (\vec\rho\,\vec\rho'_u\vec\rho'_v) \ne 0, \tag{10'} \]
\[ (\vec\rho\,\vec\rho'_u\vec\rho'_v)\frac{d^2v}{du^2} +\left(\frac{dv}{du}\right)^3(\vec\rho\,\vec\rho'_v\vec\rho''_{vv}) +\left(\frac{dv}{du}\right)^2\left[(\vec\rho\,\vec\rho'_u\vec\rho''_{vv}) +2(\vec\rho\,\vec\rho'_v\vec\rho''_{uv})\right]+ \]
\[ +\frac{dv}{du}\left[(\vec\rho\,\vec\rho'_v\vec\rho''_{uu}) +2(\vec\rho\,\vec\rho'_u\vec\rho''_{uv})\right] +(\vec\rho\,\vec\rho'_u\vec\rho''_{uu})=0, \tag{13'} \]
where \(\vec\rho=\{x;y;z\}\).
From identities (8′), (9′) we obtain the Gauss derivative formulas
\[ \vec\rho''_{uu}=\left\{\begin{matrix}11\\1\end{matrix}\right\}\vec\rho'_u+D\vec\rho,\qquad \vec\rho''_{vv}=\left\{\begin{matrix}22\\2\end{matrix}\right\}\vec\rho'_v+D''\vec\rho. \tag{14} \]
Moreover, it is known that
\[ \vec\rho''_{uv}=\left\{\begin{matrix}12\\1\end{matrix}\right\}\vec\rho'_u +\left\{\begin{matrix}12\\2\end{matrix}\right\}\vec\rho'_v+D'\vec\rho. \tag{14′} \]
If in equation (13′) we replace \(\vec{\rho}''_{uu}, \vec{\rho}''_{uv}, \vec{\rho}''_{vv}\) by their values from formulas (14), (14′), then it takes the form
\[ \frac{d^2v}{du^2} + \left(\left\{\begin{matrix}22\\2\end{matrix}\right\} -2\left\{\begin{matrix}12\\1\end{matrix}\right\}\right) \left(\frac{dv}{du}\right)^2 + \left(-\left\{\begin{matrix}11\\1\end{matrix}\right\} +2\left\{\begin{matrix}12\\2\end{matrix}\right\}\right) \left(\frac{dv}{du}\right). \tag{15} \]
This is the differential equation of geodesic lines on the sphere under the conditions
\[ \left\{\begin{matrix}22\\1\end{matrix}\right\}=0, \qquad \left\{\begin{matrix}11\\2\end{matrix}\right\}=0. \tag{16} \]
Thus, if equation (1) is nomographable, then its general solution is the solution of the differential equation of geodesics on the sphere under conditions (16).
- Let us prove the converse proposition to the one considered, i.e., that the solution of equation (15) under conditions (16) is nomographable.
The general solution of the differential equation of geodesics on the sphere has the form
\[ c_1 x(u,v)+c_2 y(u,v)+z(u,v)=0, \tag{17} \]
where \(x,y,z\) are the coordinates of a point of the sphere. Since in equation (15)
\[ \left\{\begin{matrix}11\\2\end{matrix}\right\}=0, \qquad \left\{\begin{matrix}22\\1\end{matrix}\right\}=0, \]
the equalities (14) hold. Consequently,
\[ (\vec{\rho}\,\vec{\rho}'_{u}\,\vec{\rho}''_{uu}) \equiv 0, \qquad (\vec{\rho}\,\vec{\rho}'_{v}\,\vec{\rho}''_{vv}) \equiv 0. \]
The vector \(\vec{\rho}\) is the radius vector of a point of the sphere and at the same time the unit normal vector to the sphere; therefore
\[ (\vec{\rho}\,\vec{\rho}'_{u}\,\vec{\rho}'_{v}) \not\equiv 0. \]
On the basis of conditions (8′), (9′), (10′), expression (17) can be represented in the form of a Massau determinant; consequently, equation (15) is nomographable as (1).
Received
2 III 1962
References Cited
¹ S. V. Bakhvalov, Vestn. Mosk. Univ., No. 1 (1961). ² O. D. Kellogg, Zs. f. Math. u. Phys. (1914).