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

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

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

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

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

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

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

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

myininaya Group TitleBest ResponseYou've already chosen the best response.1
and then he multiplied both sides by dt
 8 months 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? \(\color{blue}{\text{End of Quote}}\)
 8 months ago
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