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:''"Abel's formula" redirects here. For the formula on difference operators, see [[Summation by parts]].''
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In [[mathematics]], '''Abel's identity''' (also called '''Abel's differential equation identity''') is an equation that expresses the [[Wronskian]] of two solutions of a homogeneous second-order linear [[ordinary differential equation]] in terms of a coefficient of the original differential equation.
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The relation can be generalised to ''n''<sup>th</sup>-order linear ordinary differential equations. The identity is named after the [[Norway|Norwegian]] [[mathematician]] [[Niels Henrik Abel]].


Since Abel's identity relates the different [[linear independence|linearly independent]] solutions of the differential equation, it can be used to find one solution from the other. It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the [[method of variation of parameters]]. It is especially useful for equations such as [[Bessel function|Bessel's equation]] where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly.
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A generalisation to first-order systems of homogeneous linear differential equations is given by [[Liouville's formula]].
 
==Statement of Abel's identity==
Consider a [[Homogeneous differential equation|homogeneous]] linear second-order ordinary differential equation
 
: <math> y'' + p(x)y' + q(x)\,y = 0</math>
 
on an [[Interval (mathematics)|interval]] ''I'' of the [[real line]] with a [[real number|real]]- or [[complex number|complex]]-valued [[continuous function]] ''p''. Abel's identity states that the Wronskian ''W''(''y''<sub>1</sub>,''y''<sub>2</sub>) of two real- or complex-valued solutions ''y''<sub>1</sub> and ''y''<sub>2</sub> of this differential equation, that is the function defined by the [[determinant]]
 
: <math>W(y_1,y_2)(x)
=\begin{vmatrix}y_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end{vmatrix}
=y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I,</math>
 
satisfies the relation
 
: <math>W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \exp\biggl(-\int_{x_0}^x p(\xi) \,\textrm{d}\xi\biggr),\qquad x\in I,</math>
 
for every point ''x''<sub>0</sub> in ''I''.
 
===Remarks===
*In particular, the Wronskian ''W''(''y''<sub>1</sub>,''y''<sub>2</sub>) is either always the zero function or always different from zero at every point ''x'' in ''I''. In the latter case, the two solutions ''y''<sub>1</sub> and ''y''<sub>2</sub> are linearly independent (see that article about the Wronskian for a proof).
*It is not necessary to assume that the second derivatives of the solutions ''y''<sub>1</sub> and ''y''<sub>2</sub> are continuous.
*Abel's theorem is particularly useful if ''p(x)=0'', because it implies that ''W=const''.
 
===Proof of Abel's identity===
[[Derivative|Differentiating]] the Wronskian using the [[product rule]] gives (writing ''W'' for ''W''(''y''<sub>1</sub>,''y''<sub>2</sub>) and omitting the argument ''x'' for brevity)
 
: <math>
\begin{align}
W' &= y_1' y_2' + y_1 y_2'' - y_1'' y_2 - y_1' y_2' \\
& = y_1 y_2'' - y_1'' y_2.
\end{align}
</math>
 
Solving for <math>y''</math> in the original differential equation yields
 
: <math> y'' = -(py'+qy). \,</math>
 
Substituting this result into the derivative of the Wronskian function to replace the second derivatives of ''y''<sub>1</sub> and ''y''<sub>2</sub> gives
: <math>
\begin{align}
W'&= -y_1(py_2'+qy_2)+(py_1'+qy_1)y_2 \\
&= -p(y_1y_2'-y_1'y_2)\\
&= -pW.
\end{align}
</math>
 
This is a first-order linear differential equation, and it remains to show that Abel's identity gives the unique solution, which attains the value ''W''(''x''<sub>0</sub>) at ''x''<sub>0</sub>. Since the function ''p'' is continuous on ''I'', it is bounded on every closed and bounded subinterval of ''I'' and therefore integrable, hence
 
:<math>V(x)=W(x) \exp\left(\int_{x_0}^x p(\xi) \,\textrm{d}\xi\right), \qquad x\in I,</math>
 
is a well-defined function. Differentiating both sides, using the product rule, the [[chain rule]], the derivative of the [[exponential function]] and the [[fundamental theorem of calculus]], we obtain
 
:<math>V'(x)=\bigl(W'(x)+W(x)p(x)\bigr)\exp\biggl(\int_{x_0}^x p(\xi) \,\textrm{d}\xi\biggr)=0,\qquad x\in I,</math>
 
due to the differential equation for ''W''. Therefore, ''V'' has to be constant on ''I'', because otherwise we would obtain a contradiction to the [[mean value theorem]] (applied separately to the real and imaginary part in the complex-valued case). Since ''V''(''x''<sub>0</sub>) = ''W''(''x''<sub>0</sub>), Abel's identity follows by solving the definition of ''V'' for ''W''(''x'').
 
==Generalisation of Abel's identity==
Consider a homogeneous linear ''n''<sup>th</sup>-order (''n''&nbsp;≥&nbsp;1) ordinary differential equation
 
: <math>y^{(n)} + p_{n-1}(x)\,y^{(n-1)} + \cdots + p_1(x)\,y' + p_0(x)\,y = 0,</math>
 
on an interval ''I'' of the real line with a real- or complex-valued continuous function ''p''<sub>''n''&minus;1</sub>. The generalisation of Abel's identity states that the Wronskian ''W''(''y''<sub>1</sub>,…,''y<sub>n</sub>'') of ''n'' real- or complex-valued solutions ''y''<sub>1</sub>,…,''y<sub>n</sub>'' of this ''n''<sup>th</sup>-order differential equation, that is the function defined by the determinant
 
: <math>W(y_1,\ldots,y_n)(x)
=\begin{vmatrix}
y_1(x) & y_2(x) & \cdots & y_n(x)\\
y'_1(x) & y'_2(x)& \cdots & y'_n(x)\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-1)}(x) & y_2^{(n-1)}(x) & \cdots & y_n^{(n-1)}(x)
\end{vmatrix},\qquad x\in I,</math>
 
satisfies the relation
 
: <math>W(y_1,\ldots,y_n)(x)=W(y_1,\ldots,y_n)(x_0) \exp\biggl(-\int_{x_0}^x p_{n-1}(\xi) \,\textrm{d}\xi\biggr),\qquad x\in I,</math>
 
for every point ''x''<sub>0</sub> in ''I''.
 
===Direct proof===
For brevity, we write ''W'' for ''W''(''y''<sub>1</sub>,…,''y<sub>n</sub>'') and omit the argument ''x''. It suffices to show that the Wronskian solves the first-order linear differential equation
 
:<math>W'=-p_{n-1}\,W,</math>
 
because the remaining part of the proof then coincides with the one for the case ''n''&nbsp;=&nbsp;2.
 
In the case ''n''&nbsp;=&nbsp;1 we have ''W''&nbsp;=&nbsp;''y''<sub>1</sub> and the differential equation for ''W'' coincides with the one for ''y''<sub>1</sub>. Therefore, assume ''n''&nbsp;≥&nbsp;2 in the following.
 
The derivative of the Wronskian ''W'' is the derivative of the defining determinant. It follows from the [[Leibniz formula for determinants]] that this derivative can be calculated by differentiating every row separately, hence
 
: <math>\begin{align}W' & =
\begin{vmatrix}
y'_1 & y'_2 & \cdots & y'_n\\
y'_1 & y'_2 & \cdots & y'_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y'''_1 & y'''_2 & \cdots & y'''_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)}
\end{vmatrix}
+
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y'''_1 & y'''_2 & \cdots & y'''_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)}
\end{vmatrix}\\
&\qquad+\ \cdots\ +
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n\\
y'_1 & y'_2 & \cdots & y'_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-3)} & y_2^{(n-3)} & \cdots & y_n^{(n-3)}\\
y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)}\\
y_1^{(n)} & y_2^{(n)} & \cdots & y_n^{(n)}
\end{vmatrix}.\end{align}
</math>
 
However, note that every determinant from the expansion contains a pair of identical rows, except the last one. Since determinants with linearly dependent rows are equal to 0, we're only left with the last one:
 
: <math>W'=
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n\\
y'_1 & y'_2 & \cdots & y'_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)}\\
y_1^{(n)} & y_2^{(n)} & \cdots & y_n^{(n)}
\end{vmatrix}.
</math>
 
Since every ''y<sub>i</sub>'' solves the ordinary differential equation, we have
 
: <math>y_i^{(n)}  + p_{n-2}\,y_i^{(n-2)} + \cdots + p_1\,y'_i + p_0\,y_i = -p_{n-1}\,y_i^{(n-1)}</math>
 
for every ''i'' ∈ {1,...,''n''}. Hence, adding to the last row of the above determinant ''p''<sub>0</sub> times its first row, ''p''<sub>1</sub> times its second row, and so on until ''p''<sub>''n''&minus;2</sub> times its next to last row, the value of the determinant for the derivative of ''W'' is unchanged and we get
 
: <math>W'=
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n \\
y'_1 & y'_2 & \cdots & y'_n \\
\vdots & \vdots & \ddots & \vdots \\
y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)} \\
-p_{n-1}\,y_1^{(n-1)} & -p_{n-1}\,y_2^{(n-1)} & \cdots & -p_{n-1}\,y_n^{(n-1)}
\end{vmatrix}
=-p_{n-1}W.
</math>
 
===Proof using Liouville's formula===
The solutions ''y''<sub>1</sub>,…,''y<sub>n</sub>'' form the square-matrix valued solution
 
:<math>\Phi(x)=\begin{pmatrix}
y_1(x) & y_2(x) & \cdots & y_n(x)\\
y'_1(x) & y'_2(x)& \cdots & y'_n(x)\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-2)}(x) & y_2^{(n-2)}(x) & \cdots & y_n^{(n-2)}(x)\\
y_1^{(n-1)}(x) & y_2^{(n-1)}(x) & \cdots & y_n^{(n-1)}(x)
\end{pmatrix},\qquad x\in I,</math>
 
of the ''n''-dimensional first-order system of homogeneous linear differential equations
 
:<math>\begin{pmatrix}y'\\y''\\\vdots\\y^{(n-1)}\\y^{(n)}\end{pmatrix}
=\begin{pmatrix}0&1&0&\cdots&0\\
0&0&1&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&1\\
-p_0(x)&-p_1(x)&-p_2(x)&\cdots&-p_{n-1}(x)\end{pmatrix}
\begin{pmatrix}y\\y'\\\vdots\\y^{(n-2)}\\y^{(n-1)}\end{pmatrix}.</math>
 
The [[trace (linear algebra)|trace]] of this matrix is &minus;''p''<sub>''n''&minus;1</sub>(''x''), hence Abel's identity follows directly from Liouville's formula.
 
==References==
* Abel, N. H., [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0004&DMDID=dmdlog31 "Précis d'une théorie des fonctions elliptiques"] J. Reine Angew. Math, 4  (1829)  pp.&nbsp;309–348.
* Boyce, W. E. and DiPrima, R. C. (1986). ''Elementary Differential Equations and Boundary Value Problems'', 4th ed. New York: Wiley.
* {{cite book
| surname = Teschl
| given = Gerald
|authorlink=Gerald Teschl | title = Ordinary Differential Equations and Dynamical Systems
| publisher=[[American Mathematical Society]]
| place = [[Providence, Rhode Island|Providence]]
| year = 2012
| isbn = 978-0-8218-8328-0
| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
*{{MathWorld|urlname=AbelsDifferentialEquationIdentity|title=Abel's Differential Equation Identity}}
 
[[Category:Articles containing proofs]]
[[Category:Mathematical identities]]
[[Category:Ordinary differential equations]]
[[Category:Niels Henrik Abel]]

Latest revision as of 08:44, 18 July 2014

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