I found this problem in a textbook
Find a differential equation whose solution is the $n$-parameter family
$$r=x\tan(x+c)$$
The textbook says that $xr'= r + r^2 + x^2$ I'd a differential equation that works. I took the first derivative to get
$$ r' = \tan (x+c) + x\sec ^2(x+c)$$
Then I multiplied both sides by x and substituted $r$ for $x\tan (x+c)$ to get
$$xr' = r + [x\sec (x+c)]^2$$
I just don't know how the secants term in this equation seemingly becomes $r^2 + x^2$. Can anyone explain why that is or spot any errors in what I've done so far?
Never mind.
I remembered to apply a trig identity to put $(\sec x)^2$ in terms of $\tan x$ and used that to write $\tan x$ in terms of $r$ to eliminate the $c$.