i'm trying to solve this equation
\(\displaystyle U''+\frac{1}{x}U'+\frac{\lambda}{\alpha}U = 0\)
which I think is the spatial problem from separation of variables of the diffusion equation \(\displaystyle \frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}\) where \(\displaystyle T(x,t) = U(x)V(t)\)
So for \(\displaystyle U''+\frac{1}{x}U'+\frac{\lambda}{\alpha}U = 0\)
what would the characteristic equation look like for this? does the form \(\displaystyle e^{Ax}\) work?
I tried it and i got \(\displaystyle A^2+\frac{1}{x}A+\frac{\lambda}{\alpha} = 0\) but the \(\displaystyle \frac{1}{x}\) has me stuck..
\(\displaystyle U''+\frac{1}{x}U'+\frac{\lambda}{\alpha}U = 0\)
which I think is the spatial problem from separation of variables of the diffusion equation \(\displaystyle \frac{\partial^{2}T}{\partial x^{2}} + \frac{1}{x}\frac{\partial T}{\partial x} = \frac{1}{\alpha}\frac{\partial T}{\partial t}\) where \(\displaystyle T(x,t) = U(x)V(t)\)
So for \(\displaystyle U''+\frac{1}{x}U'+\frac{\lambda}{\alpha}U = 0\)
what would the characteristic equation look like for this? does the form \(\displaystyle e^{Ax}\) work?
I tried it and i got \(\displaystyle A^2+\frac{1}{x}A+\frac{\lambda}{\alpha} = 0\) but the \(\displaystyle \frac{1}{x}\) has me stuck..