I need help with 3 ODEs.

**marianne** · September 27th 2008, 12:49 PM

I'm stuck with these three ODEs:

$u'' - \frac{1}{x}u' - 4x^2u=0$
$x^2 y''(x) + xy'(x)-n^2y(x)=0$
$u''' + 2e^xu'' -u' -2e^xu=0$

Nothing I tried worked, could you please help me? A hint on which method to try would be really great.

Thank you!

**marianne** · September 27th 2008, 01:22 PM

The second one is Bessel's equation, I think. I'll Google for the solution.
If anyone has an idea for the other two..

**Jhevon** · September 27th 2008, 01:31 PM

Originally Posted by marianne

The second one is Bessel's equation, I think. I'll Google for the solution.
If anyone has an idea for the other two..

i would say the second was an Euler equation. have you considered series solutions for the other two?

**marianne** · September 27th 2008, 01:39 PM

Originally Posted by Jhevon

i would say the second was an Euler equation. have you considered series solutions for the other two?

I've found that Euler equation has the form:
$a x^2 y'' + bxy' +cy=0$ , but I don't have a constant times y, but something dependent on x..?

Edit: I thought you were talking about the first one, sorry. :-(

Also, we haven't done series solutions in class yet, but I'll try to Google it. Thank you.

**Jhevon** · September 27th 2008, 01:45 PM

Originally Posted by marianne

I've found that Euler equation has the form:
$a x^2 y'' + bxy' +cy=0$ , but I don't have a constant times y, but something dependent on x..?

Also, we haven't done series solutions in class yet, but I'll try to Google it. Thank you.

-n^2 is a constant as far as a function of x is concerned

i will think of other ways to do the problems. if you haven't done series solutions, it would be a lot to learn just to do these. so you probably aren't expected to do them that way

**marianne** · September 27th 2008, 01:51 PM

Originally Posted by Jhevon

-n^2 is a constant as far as a function of x is concerned

Yes, I know, I edited the post above, I thought you were talking about the first one. It's not an excuse, but it's late in this timezone. :-)

i will think of other ways to do the problems. if you haven't done series solutions, it would be a lot to learn just to do these. so you probably aren't expected to do them that way

Oh, thank you! And I will read about series solutions, maybe they aren't that hard to learn.

**marianne** · September 27th 2008, 01:57 PM

I solved the second one. In case someone has a similar problem, I've found this page Pauls Online Notes : Differential Equations - Euler Equations to be useful.

**NonCommAlg** · September 27th 2008, 02:55 PM

Originally Posted by marianne

$u'' - \frac{1}{x}u' - 4x^2u=0$

put $x=\sqrt{t}.$ then $u'=2\sqrt{t} \frac{du}{dt},$ and $u''=2\frac{du}{dt} + 4t \frac{d^2u}{dt^2}.$ then your differential equation becomes: $\frac{d^2u}{dt^2}-u=0,$ which is very easy to solve.

$u''' + 2e^xu'' -u' -2e^xu=0$

put $u'' - u=v.$ then your equation becomes: $v'+2e^xv=0,$ which clearly has $v=e^{-2e^x}$ as a solution. so now you need to solve $u''-u=e^{-2e^{x}}.$

**marianne** · September 28th 2008, 12:37 AM

Thank you so much!!!

**bkarpuz** · September 28th 2008, 12:57 AM

Originally Posted by marianne

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

Euler type equations have polynomial type solutions.
Just substitute $y(x)=x^{\lambda}$ and get a parabola of $\lambda$ , solve it and get two roots $\lambda_{1}$ and $\lambda_{2}$ (this is for second-order equations, for higher-order equations you get polynomial of degree $n$ ).

Then, the solution is
$ y(x)= \begin{cases} c_{1}x^{\lambda_{1}}+c_{2}x^{\lambda_{2}},&\lambda _{1}\neq\lambda_{2}\\ x^{\lambda_{1}}\big[c_{1}+c_{2}\ln(x)\big],&\lambda_{1}=\lambda_{2}, \end{cases} $
where $c_{1}$ and $c_{2}$ are arbitrary constants.

Note. Note that $\bigg(\frac{d}{dx}\bigg)^{k}$ will decrease the power of $y(x)=x^{\lambda}$ by $k$ while the factor $x^{k}$ will increase it by $k$ , hence you will get $x^{k}\bigg(\frac{d}{dx}\bigg)^{k}x^{\lambda}=\lamb da(\lambda-1)\cdots(\lambda-k+1)x^{\lambda}$ for any $k\in\mathbb{N}$ .
This is why we search solutions of polynomial type.

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