Looking for method to use in final step in heat equation problem

[mimsy57]

Homework Statement

The original problem is to solve u_t=u_xx+x with u(x,0)=0 and u(0,t)=0 by assuming there is a solution t^a*u(r), where r=x/t^b and a,b are constants

Homework Equations

The Attempt at a Solution

This is a long problem, so I'm not writing everything. Following the above, I have solved for a,b etc and reduced to the following problem, which is where I am stuck:

u''+r/2u'-3/2u=r

So my question is about what methods there are to solve this since it has non-constant coefficients, and I don't think I can use Laplace since I don't know u'(0). Does anyone know any other methods I could apply?

Thanks!

 

[kai_sikorski]

I think you have a sign error. Pretty sure it should be

\frac{3 u(r)}{2}-\frac{1}{2} r u'(r)-u''(r) = r

You can solve the homogenous equation using this

Then use variation of parameters or method of undetermined coefficients.

 

[mimsy57]

Thanks! I've been looking at this, and I think I must be using it incorrectly because I am ending up in the same place.

If I rearrange the negative to be in the same form as the page you referenced and look at the homogeneous equation:

y''+(1/2)xy'-3/2y=0

then using the labels from the page, a=1/2, b=0, α=β=0, and γ=-3/2

s is the root of the equation 4s^2+2as+α=4s^2+s=0, so s=-1/4.

Then plugging this into the equation form at the end of the page, I get u''-(x/2)u'-2u=0, which I still don't know how to solve. I thought this would make the u term go away...

Am I using this incorrectly, or am I making an arithmetic error somewhere?

Thanks again for the help!

 

[kai_sikorski]

Oh sorry, gave you the wrong link.

http://eqworld.ipmnet.ru/en/solutions/ode/ode0211.pdf

 

[mimsy57]

Okay, thanks, I really appreciate your help. This is seeming a little advanced...we haven't gone over anything like this. Could I have made a mistake earlier on? From the fact that you found the negative error, it sounds like you did it and got the same thing. Is there some way for me to determine what the first derivative is in order to use Laplace (which we HAVE learned).

 

[kai_sikorski]

I don't think so, and also due to the r multiplying the second term you would get a derivative in the laplace domain which is not ideal. The other way to solve this is using power series

http://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations

You'll get two constants A_0 and A_1. One of them should be fixed by the IC at x=0; the other can probably be determined by requiring that the series converge at infinity. So you'll want to find a constant that makes the radius of convergence for that power series infinite.

 

[mimsy57]

Thanks! This looks more like what I would know how to do. I appreciate your help!