Question

solve the differential equation. assume x,y,t>0

dy/dt=-yln(y/2). y(0)=1

Answer 1

so i think im doing this right so far but I'm not really sure can someone work through the steps with me?

Answer 2

are you taking calculus though?

Answer 3

calc 2

Answer 4

\[\frac{ dy }{ -ylny } = \frac{ 1 }{ 2 }dt\]

Answer 5

ill show the steps i have so far and can u tell me if im right or not?

Answer 6

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

Answer 7

@SithsAndGiggles

Answer 8

\[\frac{-1}{y \ln (\frac{y}{2})} dy= dt \]

Answer 9

He is doing separation of variables.

Answer 10

I think he meant to divide -yln(y/2) on both sides.

Answer 11

and then he multiplied both sides by dt

Answer 12

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?
\(\color{blue}{\text{End of Quote}}\)