Editing Linear differential equation (section)

==Non-homogeneous equation with constant coefficients==
A non-homogeneous equation of order {{mvar|n}} with constant coefficients may be written
<math display="block">y^{(n)}(x) + a_1 y^{(n-1)}(x) + \cdots + a_{n-1} y'(x)+ a_ny(x) = f(x),</math>
where {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} are real or complex numbers, {{mvar|f}} is a given function of {{mvar|x}}, and {{mvar|y}} is the unknown function (for sake of simplicity, "{{math|(''x'')}}" will be omitted in the following).

There are several methods for solving such an equation. The best method depends on the nature of the function {{mvar|f}} that makes the equation non-homogeneous. If {{mvar|f}} is a linear combination of exponential and sinusoidal functions, then the [[exponential response formula]] may be used. If, more generally, {{mvar|f}} is a linear combination of functions of the form {{math|''x''<sup>''n''</sup>''e''<sup>''ax''</sup>}}, {{math|''x''<sup>''n''</sup> cos(''ax'')}}, and {{math|''x''<sup>''n''</sup> sin(''ax'')}}, where {{mvar|n}} is a nonnegative integer, and {{mvar|a}} a constant (which need not be the same in each term), then the [[method of undetermined coefficients]] may be used. Still more general, the [[annihilator method]] applies when {{mvar|f}} satisfies a homogeneous linear differential equation, typically, a [[holonomic function]].

The most general method is the [[variation of constants]], which is presented here.

The general solution of the associated homogeneous equation 
<math display="block">y^{(n)} + a_1 y^{(n-1)} + \cdots + a_{n-1} y'+ a_ny = 0</math>
is 
<math display="block">y=u_1y_1+\cdots+ u_ny_n,</math>
where {{math|(''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>)}} is a basis of the vector space of the solutions and {{math|''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>}} are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering {{math|''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>}} as constants, they can be considered as unknown functions that have to be determined for making {{mvar|y}} a solution of the non-homogeneous equation. For this purpose, one adds the constraints
<math display="block">\begin{align}
0 &= u'_1y_1 + u'_2y_2 + \cdots+u'_ny_n \\
0 &= u'_1y'_1 + u'_2y'_2 + \cdots + u'_n y'_n \\
  &\;\;\vdots \\
0 &= u'_1y^{(n-2)}_1+u'_2y^{(n-2)}_2 + \cdots + u'_n y^{(n-2)}_n,
\end{align}</math>
which imply (by [[product rule]] and [[mathematical induction|induction]])
<math display="block">y^{(i)} = u_1 y_1^{(i)} + \cdots + u_n y_n^{(i)}</math>
for {{math|1=''i'' = 1, ..., ''n'' – 1}}, and
<math display="block">y^{(n)} = u_1 y_1^{(n)} + \cdots + u_n y_n^{(n)} +u'_1y_1^{(n-1)}+u'_2y_2^{(n-1)}+\cdots+u'_ny_n^{(n-1)}.</math>

Replacing in the original equation {{mvar|y}} and its derivatives by these expressions, and using the fact that {{math|''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>}} are solutions of the original homogeneous equation, one gets 
<math display="block">f=u'_1y_1^{(n-1)} + \cdots + u'_ny_n^{(n-1)}.</math>

This equation and the above ones with {{math|0}} as left-hand side form a system of {{mvar|n}} linear equations in {{math|''u''′<sub>1</sub>, ..., ''u''′<sub>''n''</sub>}} whose coefficients are known functions ({{mvar|f}}, the {{math|''y''{{sub|i}}}}, and their derivatives). This system can be solved by any method of [[linear algebra]]. The computation of [[antiderivative]]s gives {{math|''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>}}, and then {{math|1=''y'' = ''u''<sub>1</sub>''y''<sub>1</sub> + ⋯ + ''u''<sub>''n''</sub>''y''<sub>''n''</sub>}}.

As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.