Darboux equation

2020 Mathematics Subject Classification: Primary: 34A05 [MSN][ZBL]

An ordinary differential equation

$$\frac{\mbox{d}y}{\mbox{d}x}=\frac{P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)},$$

where $P$, $Q$ and $R$ are integral polynomials in $x$ and $y$. This equation was first studied by G. Darboux [Jo]. The Jacobi equation is a special case of the Darboux equation. Let $n$ be the highest degree of the polynomials $P$, $Q$, $R$; if the Darboux equation has $s$ known particular algebraic solutions, then if $s\geq 2+n(n+1)/2$, its general solution is found without quadratures, and if $s=1+n(n+1)/2$, an integrating factor can be found [In]. If $P$ and $Q$ are homogeneous functions of degree $m$, and $R$ is a homogeneous function of degree $k$ then, if $k=m-1$, the Darboux equation is a homogeneous differential equation; if $k\neq m-1$, the Darboux equation may be reduced to a Bernoulli equation by substituting $y=zx$.

References

[In]	E.L. Ince, "Ordinary differential equations", Dover, reprint (1956)
[Jo]	G. Darboux, "Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré" Bull. Sci. Math., 2 (1878) pp. 60–96

N.Kh. Rozov

The hyperbolic equation $$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\quad t\neq0,$$

where $\lambda(t,x)$ is a non-negative continuously-differentiable function of $x=(x_1,\ldots,x_n)$. The following uniqueness theorem is valid both for the solution of the Darboux equation and for the solution of the wave equation. If some twice continuously-differentiable solution $u(x,y)$ of the Darboux equation vanishes together with its derivative on the base of the characteristic cone lying in the plane $t=0$, it vanishes inside the entire domain bounded by this cone. The form of the characteristic cone is the same as for the wave equation. If $\lambda(t,x)=n-1>0$, the solution of the Darboux equation satisfying the initial conditions

$$u(t,x)\bigg|_{t=0}=\phi(x),\quad u_t(t,x)\bigg|_{t=0}=0,$$

with a twice continuously-differentiable function $\phi(x)$, is the function

$$u(x,t)=\frac{\Gamma(n/2)}{2\pi^{n/2}t^{n-1}}\int_{\lvert x-y\rvert=t}\phi(y)\,\mathrm{d}S_y,$$

where $\Gamma(z)$ is the gamma-function. This solution of the Darboux equation and the solution $v(x,t)$ of the wave equation satisfying the conditions

$$v(t,x)\bigg|_{t=0}=\phi(x),\quad v_t(t,x)\bigg|_{t=0}=0,$$

are connected by the relation $$u(t,x)=2\frac{\Gamma(n/2)}{\Gamma((n-1)/2)\sqrt{\pi}}\int_0^1v(t\beta,x)(1-\beta^2)^{(n-3)}/2\,\,\mathrm{d}\beta.$$

The equation was named after G. Darboux.

References

[Jo]	F. John, "Plane waves and spherical means applied to partial differential equations", Interscience (1955)