Solve the differential equation dy
dt = e−3y by separation of variables. You should obtain
an explicit formula for y as a function of t, containing a constant c (“general solution”).
For what values of t is your formula valid?

Question

anonymous · Answer

thats e^3y, sorry

anonymous · Answer

e^-3y actually

ash2326 · Answer

$$ dy/dt= e^{-3y}$$
$$dy/e^{-3y}=dt$$
$$e^{3y}dy=dt$$
integrate both sides
$$e^{3y}/3=t+c$$
$$e^{3y}=3t+c1$$
take log bot sides
$$3y=log(3t+c1)$$
$$y=(1/3)log (3t+c1)$$
3t+c1>0
or
t>-c1/3 
so the formula is valid for t>-c1/3

anonymous · Answer

$$\frac{dy}{dt} = e^{-3y}$$$$e^{3y}\frac{dy}{dt} = 1$$$$\int e^{3y}\frac{dy}{dt}dt = t + C_1$$$$\frac{1}{3}e^{3y} + C_2 = t + C_1$$$$e^{3y} = 3t + C_3$$$$y = \frac{1}{3}\ln (t+c)$$

anonymous · Answer

oops! didn't get it done in time. 
$$y = \frac{1}{3} \ln (3t+c)$$

anonymous · Answer

why are there different C's in your calculation?

turingtest · Answer

Just because the integration constant keeps changing as the formula evolves. 
C1+(another constant)=C2, for example. 
Numbering them isn't really necessary in this case, you can just call them all C. When the function is found you can apply the initial condition (if you have one) to find the final value of C.