# Writing a differential equation in power series

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• Dec 8th 2009, 06:02 PM
Toucan
Writing a differential equation in power series
Given the equation:

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

How do you write this in power series?
• Dec 8th 2009, 07:30 PM
Prove It
Quote:

Originally Posted by Toucan
Given the equation:

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

How do you write this in power series?

Assume $\displaystyle y = a + bx + cx^2 + dx^3 + \dots$.

Then $\displaystyle y' = b + 2cx + 3dx^2 + 4ex^3 + \dots$

and $\displaystyle y'' = 2c + 6dx + 12ex^2 + 20fx^3 + \dots$.

Substitute this all into the equation...
• Dec 9th 2009, 01:48 AM
chisigma
The DE equation ...

$\displaystyle x^{2}\cdot y^{''} + x\cdot y^{'} - y=0$ (1)

... belongs to a family which is called 'Euler's equation' and its particular solutions are of the form $\displaystyle x^{\alpha}$. Substituting in (1) $\displaystyle y= x^{\alpha}$ we obtain that it must be...

$\displaystyle \alpha = \pm 1$ (2)

... so that ther general solution of (1) is...

$\displaystyle y= c_{1}\cdot x + \frac{c_{2}}{x}$ (3)

... which is itself a Laurent series expansion...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$