First-order partial differential equation

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a first-order partial differential equation is a <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> that involves the first derivatives of an unknown function 
  
    
      
        u
      
    
    {\displaystyle u}
  
 of 
  
    
      
        n
        ≥
        2
      
    
    {\displaystyle n\geq 2}
  
 variables. The equation takes the form

F
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ,
        u
        ,
        
          u
          
            
              x
              
                1
              
            
          
        
        ,
        …
        
          u
          
            
              x
              
                n
              
            
          
        
        )
        =
        0
        ,
      
    
    {\displaystyle F(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0,}

using <a href="/facts/Notation_for_differentiation/PTCbwS0Y">subscript notation</a> to denote the partial derivatives of 
  
    
      
        u
      
    
    {\displaystyle u}
  
.
</p><p>Such equations arise in the construction of characteristic surfaces for <a href="/facts/Hyperbolic_partial_differential_equation/znXgGuvh">hyperbolic partial differential equations</a>, in the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>, in some <a href="/facts/Geometrical/wTQf5ffQ">geometrical</a> problems, and in simple models for gas dynamics whose solution involves the <a href="/facts/Method_of_characteristics/akP29vbW">method of characteristics</a>, e.g., the <a href="/facts/Advection_equation/3rn50V2r">advection equation</a>. If a family of solutions
of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
</p>

First-order partial differential equation open-in-new

First-order partial differential equation