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If { f1(x), f2(x) } is one set of two linearly independent solution on a<=x<=b and { g1(x), g2(x) } be another set of two linearly independent solution, then show that there exists a constant c not equal to 0 such that
W [ g1(x), g2(x) ] = c W [ f1, f2] (x)
for all a<=x<=b.
where,
W[] is wronskian, which is defined as
W [ f1(x), f2(x)]= f1 (x) * f2 ' (x) - f1' (x) * f2' (x)
(determinant involving f1, f2 and f1' , f2' )
f1 ' (x) is derivative of f1(x) i.e., d( f1(x))/dx
There is Liouville's theorem, which states that the Wronskianof two independent solutions of the ODE
satisfies
. Apply this for
and
, divide by parts, and integrate.
Are we to assume that this is a second order, linear, homogeneous differential equation? If not, it is not true. If yes, it would have been helpful to say so.Quote:
Originally Posted by niranjan
Quote:
Are we to assume that this is a second order, linear, homogeneous differential equation?
Υeap, that's the only (college/uni course maybe?) setting
where the question makes sense. (Bow)