1. differential equation

Use reduction of order to determine 2 linearly independent solutions of the differential equation

(1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1

I have done:

Assume y1 = x
Then y2 = y1 v(x)

using formula y2 = y1 int [(1/y1^2)^e -int [p dx]] dx

y2 = 1/5x^6 - x^4 + 3x^2 + 1

Can anyone please verify that I have done this correctly as I'm not too sure I've used the formula properly...

2. Originally Posted by hunkydory19
Use reduction of order to determine 2 linearly independent solutions of the differential equation

(1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1

I have done:

Assume y1 = x
Then y2 = y1 v(x)

using formula y2 = y1 int [(1/y1^2)^e -int [p dx]] dx

y2 = 1/5x^6 - x^4 + 3x^2 + 1

Can anyone please verify that I have done this correctly as I'm not too sure I've used the formula properly...

Have you substituted your solution for y2 into the DE? What do you find?

3. I found that all of the terms canceled to leave 0, so does this definitely mean that it is a solution? I thought it might just be a coincidence!

4. Originally Posted by hunkydory19
Use reduction of order to determine 2 linearly independent solutions of the differential equation

(1 - x^2)y'' + 6xy' - 6y = 0 (-1< x < 1
If $\displaystyle y_1$ is a solution the other solution is $\displaystyle y_2 = y_1\int \frac{W}{y_1} dx$. Where $\displaystyle W$ is the Wronskian.
What you do is write,
$\displaystyle y'' + \frac{6x}{1-x^2} y' - 6\frac{1}{1-x^2}y=0$
Then by Abel's theorem the Wronskain is (up to a constant) $\displaystyle W = \exp \left( - \int \frac{6x}{1-x^2} dx \right)$.