# Ordinary differential equations....

• Mar 7th 2009, 07:35 PM
niranjan
Ordinary differential equations....
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
• Jul 22nd 2009, 03:23 AM
Rebesques
There is Liouville's theorem, which states that the Wronskian $W$ of two independent solutions of the ODE $y''+ay'+by=0$ satisfies $W'(t)={\rm e}^{-\int_{t_0}^{t} a(u)du}W(t_0)$. Apply this for $W_1=W(f_1,f_2)$ and $W_2=W(g_1,g_2)$, divide by parts, and integrate.
• Jul 22nd 2009, 08:28 AM
HallsofIvy
Quote:

Originally Posted by niranjan
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

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.
• Jul 22nd 2009, 11:13 AM
Rebesques
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)