Vol. 7 No 5 - Pi Mu Epsilon
Vol. 7 No 5 - Pi Mu Epsilon
Vol. 7 No 5 - Pi Mu Epsilon
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GENERAL SOLUTION OF A GENERAL SECOND-ORDER<br />
LINEAR DIFFERENTIAL EQUATION<br />
The following theorem was discovered while attempting to find a<br />
single method for solving a general second-order linear differential<br />
equation of the form<br />
The motivation for attack on such a method stemmed from the thought that<br />
when equation (1) is associated with a first-order linear differential<br />
equation of the form<br />
- ds<br />
+ B(x)z = R(x),<br />
da<br />
then z must necessarily be of the form<br />
^- + A(x)y .<br />
d a<br />
(3<br />
FIG. 5<br />
The general forms for magic squares having magic constants<br />
(A) 4n (B) 4n+l (C) 4n+2 (D) 4n+3<br />
Working backwards now with (3) and (21, we obtain<br />
which after taking derivative and rearranging terms takes the form<br />
Comparing (1) and (4) we obtain<br />
and<br />
P(x) = A(x) t B(x)<br />
Q(x) = A(x) B (x) + A' (x).<br />
The above considerations lead us to state the following.
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