you tired using the wronskian?

Bobak

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- April 1st 2009, 10:48 AM #1

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## O.D.E. Problem

I need some help on knowing how to tackle this problem. I missed the class where we discussed this so i'm a bit clueless on how to do it .

One solution of the differential equation

(4x^2)y'' +(4x)y' + ((4x^2) -1)y = 0

is y1(x) = x^(-1/2)sinx. Find a second independent solution y2(x)

Any help is appreciated

- April 1st 2009, 10:49 AM #2

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- April 1st 2009, 11:13 AM #3

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- April 1st 2009, 11:30 AM #4

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- April 1st 2009, 11:35 AM #5

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- April 1st 2009, 08:01 PM #6
Let's write the equation as...

(1)

The general solution of (1) is...

(2)

... where and are two linearly independent sulution of (1), and two arbitrary constants.

If and are both solution of (1) then...

(3)

If we multiply the first equation by v and the second by u and take the difference we obtain...

(4)

The solution of (4) is 'very easy'...

(5)

... where is an arbitrary constant. Deviding both terms of (5) by we obtain...

(6)

Also the (6) is relatively easy to solve and we obtain...

(7)

... and from it...

(8)

The (8) allows us to obtain, if you know a solution of the (1), an other solution independent from it as follows...

(9)

In our case is...

(10)

... so that what we have to do now is to compute the integral in (9) and then find

Kind regards

- April 2nd 2009, 12:16 AM #7
Some useful comments… remembering the so called ‘Bessel equation’ …

(1)

… it is easy to verify that the proposed ODE is the Bessel equation with , the solution of which is …

(2)

With a little of patience you can find that is…

(3)

… functions that are represented here…

Setting and you can verify that is [unless the sign] …

(4)

In is interesting to note that, if and are linearly independent solution of the equation…

(5)

… with an arbitrary function of x, the relations (4) hold in any case…

Kind regards