Editing Linear differential equation (section)

{{Short description|Differential equations that are linear with respect to the unknown function and its derivatives}}
{{About|linear differential equations with one independent variable|similar equations with two or more independent variables|Partial differential equation#Linear equations of second order}}
{{Differential equations}}

In [[mathematics]], a '''linear differential equation''' is a [[differential equation]] that is [[linear equation|linear]] in the unknown function and its derivatives, so it can be written in the form 
<math display="block">a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^{(n)} = b(x)</math>
where {{nowrap|1={{math|''a''<sub>0</sub>(''x'')}}, ..., {{math|''a''<sub>''n''</sub>(''x'')}}}} and {{math|''b''(''x'')}} are arbitrary [[differentiable function]]s that do not need to be linear, and {{math|''y''′, ..., ''y''<sup>(''n'')</sup> }} are the successive derivatives of an unknown function {{mvar|y}} of the variable {{mvar|x}}.

Such an equation is an [[ordinary differential equation]] (ODE). A ''linear differential equation'' may also be a linear [[partial differential equation]] (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are [[partial derivative]]s.