# B Recast a differential equation with a change of variable

1. Mar 22, 2016

### Happiness

I obtained an extra factor of $\frac{1}{l^2}$ in the first term on the LHS of (3.34).

From (3.33),

LHS $=u^2\frac{d}{d\theta}(\frac{u^2}{m}\frac{du}{d\theta}\frac{dr}{du})-\frac{l^2u^3}{m}$
$=u^2\frac{d}{d\theta}(-\frac{1}{m}\frac{du}{d\theta})-\frac{l^2u^3}{m}$
$=-\frac{u^2}{m}\frac{d^2u}{d\theta^2}-\frac{l^2u^3}{m}$

Multiplying throughout by $-\frac{m}{l^2u^2}$, we have
LHS $=\frac{1}{l^2}\frac{d^2u}{d\theta^2}+u$
which differs from (3.34).

What's wrong?

EDIT: I found the mistake. The text has a typo. The "1"s in (3.33) are actually "$l$"s.

Last edited: Mar 22, 2016
2. Mar 22, 2016

### BvU

Well done !

3. Mar 22, 2016

### Ssnow

yes, this was the error ...