# Thread: homogenous solution of diff. ekv

1. ## homogenous solution of diff. ekv

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

Thanks.

2. 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

Thanks.
For the homogeneous problem

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

or

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

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

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

although I think you have a problem with your IC.

3. How did you go from with m=1.1 to the solution
?

4. Originally Posted by danielel
How did you go from with m=1.1 to the solution
?
Since $\displaystyle m = -1,\;\; m = 1$ then two solutions are $\displaystyle u = r^1,\;\;\text{and}\;\; u = r^{-1}$ and since the ODE is linear then any linear combination is also a solution giving $\displaystyle u = c_1 r + \frac{c_2}{r}$