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Difference between revisions of "Darboux equation"

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An ordinary differential equation
 
An ordinary differential equation
  
 
$$\frac{\mbox{d}y}{\mbox{d}x}=\frac{P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)},$$
 
$$\frac{\mbox{d}y}{\mbox{d}x}=\frac{P(x,y)+yR(x,y)}{Q(x,y)+xR(x,y)},$$
  
where $P$, $Q$ and $R$ are integral polynomials in $x$ and $y$. This equation was first studied by G. Darboux [[#References|[1]]]. The [[Jacobi equation|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 [[#References|[2]]]. 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|Bernoulli equation]] by substituting $y=zx$.
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where $P$, $Q$ and $R$ are integral polynomials in $x$ and $y$. This equation was first studied by G. Darboux {{Cite|Jo}}. The [[Jacobi equation|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 {{Cite|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|Bernoulli equation]] by substituting $y=zx$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Darboux,  "Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré"  ''Bull. Sci. Math.'' , '''2'''  (1878)  pp. 60–96</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>
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|valign="top"|{{Ref|In}}||valign="top"|  E.L. Ince,  "Ordinary differential equations", Dover, reprint  (1956)
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|valign="top"|{{Ref|Jo}}||valign="top"| G. Darboux,  "Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré"  ''Bull. Sci. Math.'', '''2'''  (1878)  pp. 60–96
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''N.Kh. Rozov''
 
''N.Kh. Rozov''
  
 
The hyperbolic equation
 
The hyperbolic equation
$$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\,\,\,t\neq0,$$
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$$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\quad t\neq0,$$
  
  
 
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|wave equation]]. If $\lambda(t,x)=n-1>0$, the solution of the Darboux equation satisfying the initial conditions
 
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|wave equation]]. If $\lambda(t,x)=n-1>0$, the solution of the Darboux equation satisfying the initial conditions
  
$$u(t,x)\bigg|_{t=0}=\phi(x),\,\,\,u_t(t,x)\bigg|_{t=0}=0,$$
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$$u(t,x)\bigg|_{t=0}=\phi(x),\quad u_t(t,x)\bigg|_{t=0}=0,$$
  
 
with a twice continuously-differentiable function $\phi(x)$, is the function
 
with a twice continuously-differentiable function $\phi(x)$, is the function
  
$$u(x,t)=\frac{\Gamma(n/2)}{2\pi^{n/2}t^{n-1}}\int_{\lvert x-y\rvert=t}\phi(y)\,\mbox{d}S_y,$$
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$$u(x,t)=\frac{\Gamma(n/2)}{2\pi^{n/2}t^{n-1}}\int_{\lvert x-y\rvert=t}\phi(y)\,\mathrm{d}S_y,$$
  
 
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
 
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
  
$$v(t,x)\bigg|_{t=0}=\phi(x),\,\,\,v_t(t,x)\bigg|_{t=0}=0,$$
+
$$v(t,x)\bigg|_{t=0}=\phi(x),\quad v_t(t,x)\bigg|_{t=0}=0,$$
  
 
are connected by the relation
 
are connected by the relation
$$u(t,x)=2\frac{\Gamma(n/2)}{\Gamma((n-1)/2)\sqrt{\pi}}\int_0^1v(t\beta,x)(1-\beta^2)^{(n-3)}/2\,\,\mbox{d}\beta.$$
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$$u(t,x)=2\frac{\Gamma(n/2)}{\Gamma((n-1)/2)\sqrt{\pi}}\int_0^1v(t\beta,x)(1-\beta^2)^{(n-3)}/2\,\,\mathrm{d}\beta.$$
  
  
 
The equation was named after G. Darboux.
 
The equation was named after G. Darboux.
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. John,  "Plane waves and spherical means applied to partial differential equations" , Interscience  (1955)</TD></TR></table>
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|valign="top"|{{Ref|Jo}}||valign="top"| F. John,  "Plane waves and spherical means applied to partial differential equations", Interscience  (1955)
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Latest revision as of 11:53, 20 April 2012

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)
How to Cite This Entry:
Darboux equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_equation&oldid=24848
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article