# Partial differential equations , rearranging and spotting

1. Aug 27, 2011

### mohsin031211

I have been given an equation : r^2 ( d^2R/dr^2) + 2r(dR/dr) - lambda*R = 0

It says to assume R~ r^β

Then i can't seem to spot how from that information we can produce this equation:

β(β − 1) rβ + 2β rβ − λ rβ = 0

Any help would be appreciated, thanks.

2. Aug 27, 2011

### Hootenanny

Staff Emeritus
Simply substitute $R=r^\beta$ into the differential equation.

3. Aug 27, 2011

### HallsofIvy

Staff Emeritus
That is, by the way, an "Euler-Lagrange" type equation. Each derivative is multiplied by a power of x equal to the order of the derivative. The substitution t= ln(r) changes it to a "constant coefficients" problem. You should remember that for such an equation we "try" a solution of the form $e^{\beta t}$ (although we then find that there are other solutions). With t= ln r, that becomes $e^{\beta ln(r)}= e^{ln r^\beta}= r^\beta$.

4. Aug 27, 2011

### stallionx

Set R= constant * r**beta so dR=constant * whatever

They are proportional
R=k*r--beta , all k's ( constants cancel )

5. Aug 27, 2011

### Hootenanny

Staff Emeritus
Why the constant?

6. Aug 27, 2011

### stallionx

Well because I thought tilde is for (constant) linear proportionality.

7. Aug 29, 2011

### acnes

I have been given an equation