Linear differential equation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a linear differential equation is a <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> that is <a href="/facts/Linear_equation/JfCgALwf">linear</a> in the unknown function and its derivatives, so it can be written in the form

a
          
            0
          
        
        (
        x
        )
        y
        +
        
          a
          
            1
          
        
        (
        x
        )
        
          y
          ′
        
        +
        
          a
          
            2
          
        
        (
        x
        )
        
          y
          ″
        
        ⋯
        +
        
          a
          
            n
          
        
        (
        x
        )
        
          y
          
            (
            n
            )
          
        
        =
        b
        (
        x
        )
      
    
    {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}

where a0(x), ..., an(x) and b(x) are arbitrary <a href="/facts/Differentiable_function/jQqTgLk1">differentiable functions</a> that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.
Such an equation is an <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> (ODE). A linear differential equation may also be a linear <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a>.

Linear differential equation open-in-new

Linear differential equation