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Math Help - technical solution for this 2nd order de..

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by p00ndawg View Post
    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.
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