v"t + v' = 0
my professor solved this by guessing, but i was wondering how would I solve it the technical way?
I am guessing that v is a function of t if so
let $\displaystyle u=v' \implies u'=v''$
subbing in we get
$\displaystyle t u'+u=0 \iff t\frac{du}{dt}=-u \iff \frac{du}{u}=-\frac{du}{t}$
integrating both sides gives
$\displaystyle \ln|u|=-\ln|t|+\ln|c| \iff ln|u|=\ln \left( \frac{c}{t}\right)$
$\displaystyle u=\frac{c}{t}$ but u=v' so we get
$\displaystyle \frac{dv}{dt}=\frac{c}{t} \iff dv=\frac{c}{t}dt \implies v=c\ln|t|+d$
I hope this helps.