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

1. megannicole51

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

2. ddog1437

are you taking calculus though?

3. megannicole51

calc 2

4. megannicole51

$\frac{ dy }{ -ylny } = \frac{ 1 }{ 2 }dt$

5. megannicole51

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

6. megannicole51

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

7. megannicole51

@SithsAndGiggles

8. myininaya

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

9. myininaya

He is doing separation of variables.

10. myininaya

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

11. myininaya

and then he multiplied both sides by dt

12. SithsAndGiggles

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? $$\color{blue}{\text{End of Quote}}$$