Given a homogeneous linear Second-Order Ordinary Differential Equation,
![\begin{displaymath}
y''+P(x)y'+Q(x)y=0,
\end{displaymath}](a_48.gif) |
(1) |
call the two linearly independent solutions
and
. Then
![\begin{displaymath}
y''_1+P(x)y'_1+Q(x)y_1=0
\end{displaymath}](a_51.gif) |
(2) |
![\begin{displaymath}
y''_2+P(x)y'_2+Q(x)y_2=0.
\end{displaymath}](a_52.gif) |
(3) |
Now, take
(3) minus
(2),
![\begin{displaymath}
y_1[y''_2+P(x)y'_2+Q(x)y_2]-y_2[y''_1+P(x)y'_1+Q(x)y_1]=0
\end{displaymath}](a_55.gif) |
(4) |
![\begin{displaymath}
(y_1y''_2-y_2y''_1)+P(y_1y'_2-y'_1y_2)+Q(y_1y_2-y_1y_2)=0
\end{displaymath}](a_56.gif) |
(5) |
![\begin{displaymath}
(y_1y''_2-y_2y''_1)+P(y_1y'_2-y'_1y_2)=0.
\end{displaymath}](a_57.gif) |
(6) |
Now, use the definition of the Wronskian and take its Derivative,
Plugging
and
into (6) gives
![\begin{displaymath}
W'+PW = 0.
\end{displaymath}](a_66.gif) |
(9) |
This can be rearranged to yield
![\begin{displaymath}
{dW\over W} = -P(x)\,dx
\end{displaymath}](a_67.gif) |
(10) |
which can then be directly integrated to
![\begin{displaymath}
\ln\left[{W(x)\over W_0}\right]= - \int P(x)\,dx,
\end{displaymath}](a_68.gif) |
(11) |
where
is the Natural Logarithm. Exponentiation then yields Abel's identity
![\begin{displaymath}
W(x) = W_0e^{-\int P(x)\,dx},
\end{displaymath}](a_70.gif) |
(12) |
where
is a constant of integration.
See also Ordinary Differential Equation--Second-Order
References
Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed.
New York: Wiley, pp. 118, 262, 277, and 355, 1986.
© 1996-9 Eric W. Weisstein
1999-05-25