# technical solution for this 2nd order de..

• Mar 11th 2009, 07:31 PM
p00ndawg
technical solution for this 2nd order de..
v"t + v' = 0

my professor solved this by guessing, but i was wondering how would I solve it the technical way?
• Mar 11th 2009, 07:41 PM
TheEmptySet
Quote:

Originally Posted by p00ndawg
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 $u=v' \implies u'=v''$

subbing in we get

$t u'+u=0 \iff t\frac{du}{dt}=-u \iff \frac{du}{u}=-\frac{du}{t}$

integrating both sides gives

$\ln|u|=-\ln|t|+\ln|c| \iff ln|u|=\ln \left( \frac{c}{t}\right)$

$u=\frac{c}{t}$ but u=v' so we get

$\frac{dv}{dt}=\frac{c}{t} \iff dv=\frac{c}{t}dt \implies v=c\ln|t|+d$

I hope this helps.