# solve this Differential equation by indicial equation

• May 18th 2008, 06:51 PM
szpengchao
solve this Differential equation by indicial equation
• May 18th 2008, 06:55 PM
mr fantastic
Quote:
Given the form of the DE, I would assume that the first part requires you to find a solution of the form $\displaystyle Ax^n$.

Where are you stuck in the second part?
• May 18th 2008, 07:02 PM
szpengchao
problem
i have problem with the first part
• May 18th 2008, 07:11 PM
TheEmptySet
Quote:
Hint:
$\displaystyle z=\ln(x)$

Use the chain rule

$\displaystyle \frac{dy}{dx}=\frac{dy}{dz}\frac{dz}{dx}=\frac{1}{ x}\frac{dy}{dz}$

Now we need the 2nd derivative using the product and chain rule

$\displaystyle \frac{d^2y}{dx^2}=-\frac{1}{x^2}\frac{dy}{dz}+\frac{1}{x}\left( \frac{d^2y}{dz^2} \cdot \frac{1}{x}\right)$

from here sub into the equation

Good luck
• May 18th 2008, 07:15 PM
szpengchao
no
no...i mean the 1st question...not the second
• May 18th 2008, 07:23 PM
TheEmptySet
Quote:

Originally Posted by szpengchao
no...i mean the 1st question...not the second

Are you talking about using the method of frobenius to obtain series solutions to the ODE?
• May 18th 2008, 07:24 PM
szpengchao
yeah
yeah.... i have problem with choosing point for this particular question...
would you like to post ur steps?
• May 18th 2008, 07:45 PM
szpengchao
choosing ordinary point
well, x=0 is a regular singular point, right?
then , i set y= sigma( A_n * x^(n+b)) into the equation. then i got something like:

sigma( A_n * (n^2+1) = 0) well, this means A_n = 0 for all n...then how can i continue?
• May 18th 2008, 08:34 PM
TheEmptySet
Quote:

Originally Posted by szpengchao
well, x=0 is a regular singular point, right?
then , i set y= sigma( A_n * x^(n+b)) into the equation. then i got something like:

sigma( A_n * (n^2+1) = 0) well, this means A_n = 0 for all n...then how can i continue?

$\displaystyle y=\sum_{n=0}^{\infty}c_nx^{n+r}$

$\displaystyle y'=\sum_{n=0}^{\infty}c_n(n+r)x^{n+r-1}$

$\displaystyle y''=\sum_{n=2}^{\infty}c_n(n+r)(n+r-1)x^{n+r-2}$

$\displaystyle x^2y''+xy'+y=0$

$\displaystyle x^2\sum_{n=0}^{\infty}c_n(n+r)(n+r-1)x^{n+r-2}+x\sum_{n=0}^{\infty}c_n(n+r)x^{n+r-1}+ \sum_{n=0}^{\infty}c_nx^{n+r}=0$

$\displaystyle \sum_{n=0}^{\infty}c_n(n+r)(n+r-1)x^{n+r}+\sum_{n=0}^{\infty}c_n(n+r)x^{n+r}+ \sum_{n=0}^{\infty}c_nx^{n+r}=0$

$\displaystyle c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)(n+r-1)+(n+r)+ 1 \right)=0$

$\displaystyle c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)[(n+r-1)+1]+ 1 \right)=0$

$\displaystyle c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)^2+ 1 \right)=0$

I think this is where you are stuck.
• May 18th 2008, 08:36 PM
szpengchao
yup
yup...then? how can that be zero?
• May 18th 2008, 08:49 PM
TheEmptySet
Quote:

Originally Posted by szpengchao
yup...then? how can that be zero?

I don't know besides the trivial solution. Maybe someone can find my mistake.

(Crying)
• May 18th 2008, 10:13 PM
mr fantastic
Quote:

Originally Posted by TheEmptySet
I don't know besides the trivial solution. Maybe someone can find my mistake.

(Crying)

Using the substitution given in the second part, I get $\displaystyle y = A \sin (\ln x) + B \cos (\ln x)$ ......
• May 19th 2008, 01:21 AM
mr fantastic
Quote:

Originally Posted by mr fantastic
Given the form of the DE, I would assume that the first part requires you to find a solution of the form $\displaystyle Ax^n$.

Where are you stuck in the second part?

Do this and you get $\displaystyle n^2 + 1 = 0 \Rightarrow n = \pm i$.

Therefore

$\displaystyle y = A_1 x^i + A_2 x^{-i} = A_1 e^{i \ln x} + A_2 e^{-i \ln x} = B_1 \cos (\ln x) + B_2 \sin (\ln x)$

by the magic of Euler's equation and the usual identity.
• May 19th 2008, 08:56 AM
TheEmptySet
Thanks Mr. Fantastic,

I forgot that $\displaystyle c_0 \ne 0$ in this method so

$\displaystyle c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)^2+ 1 \right)=0 \iff c_0x^{r}(r^2+1)+c_nx^{n+r}\left(\sum_{n=1}^{\infty }(n+r)^2+ 1 \right)=0$

so $\displaystyle c_n=0, \forall n \ge 1$

$\displaystyle r^2=1 \iff r=\pm i$

Then the rest follows.

Thanks again!(Clapping)