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Lengalicious
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Homework Statement
See attachment (stuck with part b at the moment)
Homework Equations
The Attempt at a Solution
[tex]\phi=D(x)T(t)[/tex]
so
[tex](1+bx)D''(x)T(t)-D(x)T''(t)=0[/tex]
[tex](1+bx)\frac{D''(x)}{D(x)}=\frac{T''(t)}{T(t)}[/tex]
let
[tex]\frac{T''}{T}=\sigma[/tex] (1)
use trial solution [tex]T=be^{rt}[/tex]
subbing into (1) and solve for r.
[tex]r=\pm\sqrt{\sigma}[/tex]
use same trial solution and repeat steps for
[tex](1+bx)\frac{D''}{D}=\sigma[/tex]
[tex]r=\pm\sqrt{\frac{\sigma}{1+bx}}[/tex]
from principle of superposition
[tex]D(x)=a_1e^{\sqrt{\frac{\sigma}{1+bx}}x}+a_2e^{-\sqrt{\frac{\sigma}{1+bx}}x}[/tex]
[tex]T(t)=b_1e^{\sqrt{\sigma}t}+b_2e^{-\sqrt{\sigma}t}[/tex]
Then I get confused with boundary conditions can someone let me know if I am on the right lines so far and give me any advice for proceeding?
Thanks
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