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VinnyCee
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Homework Statement
Apply the method of Frobenius to find the roots of the indicial equation to show that [itex]c_1\,=\,c_2\,=\,0[/itex].
The equation in question is a 2nd order DE that was https://www.physicsforums.com/showthread.php?t=177492".
[tex]z\,\frac{d^2T}{dx^2}\,+\,\frac{z}{x}\,\frac{dT}{dx}\,-\,z\,y\,=\,0[/tex]
If you look at the thread where this equation was derived, we assume that [itex]z\,=\,x^2[/itex].
[tex]x^2\,\frac{d^2T}{dx^2}\,+\,\frac{x^2}{x}\,\frac{dT}{dx}\,-\,x^2\,y\,=\,0[/tex]
Homework Equations
http://en.wikipedia.org/wiki/Frobenius_method" [Broken]
The Attempt at a Solution
There is a regular, singular point at [itex]x_0\,=\,0[/itex]. We seek a solution of the form
[tex]\sum_0^\infty\,a_n\,x^{n\,+\,r}[/tex]
Differentiating that sum twice and substituting into the steady-state heat balance equation above and bringing the x's into the sums
[tex]\sum_0^\infty\,(n\,+\,r)(n\,+\,r\,+1)\,a_n\,x^{n\,+\,r}\,+\,\sum_0^\infty\,(n\,+\,r)\,a_n\,x^{n\,+\,r\,-1}\,-\,\sum_0^\infty\,a_n\,x^{n\,+\,r\,+\,2}\,+\,T_a\,x^2\,=\,0[/tex]
Note the last term, it arises from using [itex]y\,=\,T\,-\,T_a[/itex] from the https://www.physicsforums.com/showthread.php?t=177492". I really don't know what to do with it, I don't think it is constant, but how do I combine into sums and how do I deal with the constant [itex]T_a[/itex]? I am going to eliminate it, but don't know why!
Now, changing the indicies to combine the summations
[tex]\sum_0^\infty\,\left[(n\,+\,r)(n\,+\,r\,+\,1)\,a_n\,+\,(n\,+\,r\,+1)\,a_{n\,+\,1}\,-\,a_{n\,-\,2}\right]\,x^{n\,+\,r}\,=\,0[/tex]
Now, I get the indicial equation
[tex](r^2\,-\,r)\,a_0\,+\,(r\,+\,1)\,a_1\,=\,0[/tex]
[tex]r^2\,-\,r\,=\,0\,\longrightarrow\,r^2\,-\,r\,=\,0\,\longrightarrow\,r\,=\,0,\,1[/tex]
But the roots are both supposed to be zero, what did I do wrong?
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