# Solving a differential equation using power series

• May 18th 2011, 04:11 AM
garunas
Solving a differential equation using power series
Let $\displaystyle \rho \in \mathbb{R}$ and let

$\displaystyle y(x) = x^\rho\left(1 + \sum_{n=1}^\infty a_{n}x^n\right)\hspace{10mm}(1)$

be a solution to the differential equation:

$\displaystyle y''(x) + \left(1 + \frac {1}{x} \right)y'(x) - \frac{3}{x^2} y(x) = 0\hspace{10mm}(2)$

The first question asks: "What is the set of possible values for $\displaystyle \rho$?" For which i have found that
$\displaystyle \rho = -\sqrt{3}$ and $\displaystyle \rho = \sqrt{3}$

The second question (the one I'm stuck on) asks: "By substituting (1) into (2) for each value of $\displaystyle \rho$ found in the previous question determine the corresponding coefficient $\displaystyle a_{1}$ in (1). I have somewhat done this and got this equation:

$\displaystyle x^{\rho - 2}\left(\rho^2 - 3\right) \hspace{3mm}+\hspace{3mm} \rho x^{\rho - 1} \hspace{3mm}+\hspace{3mm} \sum_{n=1}^\infty ((n+\rho)^2 - 3)a_{n}x^{n+\rho - 2} \hspace{3mm}+\hspace{3mm} \sum_{n=1}^\infty \left(n + \rho)a_{n}x^{n + \rho -1} = 0$

Where do I go from here? Is my equation even right in the first place?
• May 23rd 2011, 05:41 PM
Sudharaka
Quote:

Originally Posted by garunas
Let $\displaystyle \rho \in \mathbb{R}$ and let

$\displaystyle y(x) = x^\rho\left(1 + \sum_{n=1}^\infty a_{n}x^n\right)\hspace{10mm}(1)$

be a solution to the differential equation:

$\displaystyle y''(x) + \left(1 + \frac {1}{x} \right)y'(x) - \frac{3}{x^2} y(x) = 0\hspace{10mm}(2)$

The first question asks: "What is the set of possible values for $\displaystyle \rho$?" For which i have found that
$\displaystyle \rho = -\sqrt{3}$ and $\displaystyle \rho = \sqrt{3}$

The second question (the one I'm stuck on) asks: "By substituting (1) into (2) for each value of $\displaystyle \rho$ found in the previous question determine the corresponding coefficient $\displaystyle a_{1}$ in (1). I have somewhat done this and got this equation:

$\displaystyle x^{\rho - 2}\left(\rho^2 - 3\right) \hspace{3mm}+\hspace{3mm} \rho x^{\rho - 1} \hspace{3mm}+\hspace{3mm} \sum_{n=1}^\infty ((n+\rho)^2 - 3)a_{n}x^{n+\rho - 2} \hspace{3mm}+\hspace{3mm} \sum_{n=1}^\infty \left(n + \rho)a_{n}x^{n + \rho -1} = 0$

Where do I go from here? Is my equation even right in the first place?

Dear garunas,

I hope your equation is correct. By rearranging that you could obtain,

$\displaystyle x^{\rho-2}(\rho^2-3)+\left[\rho+\left((1+\rho)^2-3\right)a_1\right]x^{\rho-1}+\sum_{n=1}^{\infty}\left[\left((n+1+\rho)^2-3\right)a_{n+1}+(n+\rho)a_n\right]x^{n+\rho-1}=0$

Now by equating coefficients of similar powers of x to zero,

$\displaystyle \rho^2-3=0\rightarrow \rho=\pm\sqrt{3}$

$\displaystyle \rho+\left((1+\rho)^2-3\right)a_1=0\Rightarrow{a_1=\frac{-\rho}{\rho^2+2\rho-2}$

When $\displaystyle \rho=\sqrt{3}$,

$\displaystyle a_1=\frac{-\sqrt{3}}{2\sqrt{3}+1}$

When $\displaystyle \rho=-\sqrt{3}$,

$\displaystyle a_1=\frac{\sqrt{3}}{1-2\sqrt{3}}$

Hope you understood.