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megannicole51
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solve the differential equation. assume x,y,t>0
dy/dt=yln(y/2). y(0)=1
 one year ago
 one year ago
megannicole51 Group Title
solve the differential equation. assume x,y,t>0 dy/dt=yln(y/2). y(0)=1
 one year ago
 one year ago

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megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
so i think im doing this right so far but I'm not really sure can someone work through the steps with me?
 one year ago

ddog1437 Group TitleBest ResponseYou've already chosen the best response.0
are you taking calculus though?
 one year ago

megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
calc 2
 one year ago

megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
\[\frac{ dy }{ ylny } = \frac{ 1 }{ 2 }dt\]
 one year ago

megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
ill show the steps i have so far and can u tell me if im right or not?
 one year ago

megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
ive never seen it done like that actually....my professor hasnt really done any examples so i am trying to figure it out from our book which isnt helping either
 one year ago

megannicole51 Group TitleBest ResponseYou've already chosen the best response.0
@SithsAndGiggles
 one year ago

myininaya Group TitleBest ResponseYou've already chosen the best response.1
\[\frac{1}{y \ln (\frac{y}{2})} dy= dt \]
 one year ago

myininaya Group TitleBest ResponseYou've already chosen the best response.1
He is doing separation of variables.
 one year ago

myininaya Group TitleBest ResponseYou've already chosen the best response.1
I think he meant to divide yln(y/2) on both sides.
 one year ago

myininaya Group TitleBest ResponseYou've already chosen the best response.1
and then he multiplied both sides by dt
 one year ago

SithsAndGiggles Group TitleBest ResponseYou've already chosen the best response.1
Thanks to @myininaya for pointing out my mistake: \(\color{blue}{\text{Originally Posted by}}\) @SithsAndGiggles \[\frac{dy}{dt}=y\ln\left(\frac{y}{2}\right)\\ \frac{1}{y\ln\left(\frac{y}{2}\right)}~dy=dt\] Integrate both sides: \[\int\frac{1}{y\ln\left(\frac{y}{2}\right)}~dy=\int dt\] First, a substitution: \(u=\dfrac{y}{2}\), or \(2u=y\), so that \(2du=dy\): \[\int\frac{1}{2u\ln u}~(2~du)=\int dt\\ \int\frac{1}{u \ln u}~du=\int dt\] Got everything so far? \(\color{blue}{\text{End of Quote}}\)
 one year ago
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