Definition:Differential Equation/Linear

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A linear differential equation is a differential equation where all dependent variables and their derivatives appear to the first power.

Neither are products of dependent variables allowed.

Hence a linear differential equation is of the form:

$\map {P_0} x y + \map {P_1} x \dfrac {\d y} {\d x} + \map {P_2} x \dfrac {\d^2 y} {\d x^2} + \cdots + \map {P_n} x \dfrac {\d^n y} {\d x^n} = \map Q x$

where $P_0, P_1, \ldots, P_n, Q$ are functions of $x$.


Arbitrary Example

The following is an example of a linear differential equation:

$x \dfrac {\d y} {\d x} + y = \sin x$

Also see

  • Results about linear differential equations can be found here.