# Formula that gives all possible solutions

• Apr 12th 2012, 07:09 PM
reliance
Formula that gives all possible solutions
Hi there,
I'm working on a problem that asks me to find a formula that'll give all possible solutions to the equation V'' + (2/3)V'(r) = 0. I'm not certain where to begin. I believe that solving for V'(r) would be a nice first step, but I don't know how to go about this either. ANy guidance would be greatly appreciated. Thank you!
• Apr 12th 2012, 07:43 PM
pickslides
Re: Formula that gives all possible solutions
Look at this link under "Second Order Differential equations", the first example will show you how to proceed.

First and Second Order Differential Equations
• Apr 12th 2012, 08:57 PM
reliance
Re: Formula that gives all possible solutions
Thank you for the link pickslides.

From what I've gotten, the solution should be something akin to y = c(sub 1)e^(r(sub1)x) + c(sub 2)e^(r(sub 2)x). The solution to r^2 - (2/r)r = 0 is plus or minus square root of 2. The solution, then, appears to be y = c(sub 1)e^(sqrt(2)x) + c(sub 2)e^(-sqrt(2)x). Is this correct? Thank you once again for your assistance.
• Apr 12th 2012, 09:11 PM
pickslides
Re: Formula that gives all possible solutions
$\displaystyle \lambda^2+\frac{2}{3}\lambda = 0$

$\displaystyle \lambda\left( \lambda+\frac{2}{3}\right) = 0$

By the null factor law $\lambda = 0, \frac{-2}{3}$

Now the solution to $V(r)$

Now $\displaystyle V(r) = C_1e^{\frac{-2r}{3}}+C_2e^{0}$
• Apr 12th 2012, 09:18 PM
reliance
Re: Formula that gives all possible solutions
pickslides, I'm so sorry. I typed the coefficient (2/3) in the original problem when the problem states (2/r). I typed my attempt at a solution with regard to that. I apologize sincerely for my mistake. You've been very kind with your explanation, and I thank you.
• Apr 12th 2012, 09:55 PM
pickslides
Re: Formula that gives all possible solutions
This makes things more difficult.

Have a look at this link

solve y&#39;&#39;&#43;&#40;2&#47;x&#41;y&#39; &#61;0 - Wolfram|Alpha
• Apr 13th 2012, 09:45 AM
HallsofIvy
Re: Formula that gives all possible solutions
Quote:

Originally Posted by reliance
Hi there,
I'm working on a problem that asks me to find a formula that'll give all possible solutions to the equation V'' + (2/3)V'(r) = 0. I'm not certain where to begin. I believe that solving for V'(r) would be a nice first step, but I don't know how to go about this either. ANy guidance would be greatly appreciated. Thank you!

The equation is V''+ (2/r)V'= 0. Since V itself does not appear in the equation, let U= V' so the equation becomes U'+ (2/r)U= 0. That is a separable first order equation. We can write dU/dr= -(2/r)U and then dU/U= -(2 dr/r). Integrating both sides, $ln(U)= -2ln(r)+ C= ln(r^{-2}+ C$ so that $U= C' r^{-2}$ where $C'= e^C$. Going back to V we have $\frac{dV}{dr}= C'r^{-2}$ and that's an easy integral.