homogenous solution of diff. ekv

**danielel** · February 20th 2009, 01:51 AM

Hello

I have a linear differential ekvation
d2u/dr2 + 1/r*du/dr -u/r^2=-(1-v)*pw
I want to have the homogenous solution to this,
I know I should put the right side equal to zero.
But after that my mind is blank...
I think that u(0)=0
Please help

Thanks.

**Danny** · February 20th 2009, 04:56 AM

Originally Posted by danielel

Hello

I have a linear differential ekvation
d2u/dr2 + 1/r*du/dr -u/r^2=-(1-v)*pw
I want to have the homogenous solution to this,
I know I should put the right side equal to zero.
But after that my mind is blank...
I think that u(0)=0
Please help

Thanks.

For the homogeneous problem

$\frac{d^2u}{dr^2} + \frac{1}{r} \frac{du}{dr} - \frac{u}{r^2} = 0$

or

$r^2 \frac{d^2u}{dr^2} + r \frac{du}{dr} - u = 0$

if you seek solutions of the form $u = r^m$ then we find that m satisfies $m(m-1) + m - 1 = 0$ (typically called the characteristic equation) and this gives $m = -1, 1$ and the solution

$u =c_1 r + \frac{c_2}{r}$

although I think you have a problem with your IC.

**danielel** · February 20th 2009, 05:05 AM

How did you go from

with m=1.1 to the solution

?

**Danny** · February 20th 2009, 05:36 AM

Originally Posted by danielel

How did you go from

with m=1.1 to the solution

?

Since $m = -1,\;\; m = 1$ then two solutions are $u = r^1,\;\;\text{and}\;\; u = r^{-1}$ and since the ODE is linear then any linear combination is also a solution giving $u = c_1 r + \frac{c_2}{r}$

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