View Full Version : A level integration
albear
Apr 28, 2007, 08:46 AM
right OK, the mass M at time t of the leaves of a certain plant varies according to the differential equation
dM/dt=M-M^2
a given that at time t=0, M=0.5 find an expression for M in terms of t
here is my wrong answer
(intregral) 1/M*dM=(integral) 1-M*dt
lnM=t-Mt
M=e^t * e^-mt
(the right answer is M=(e^t)/1+e^t)
how does it get to this?
galactus
Apr 28, 2007, 10:37 AM
Since you gave this an earnest effort, I will show you how they got that.
\frac{dM}{dt}=M-M^{2}
Separate variables:
\frac{dM}{M-M^{2}}=dt
Integrate:
\int\frac{dM}{M-M^{2}}=\int{dt}
ln(\frac{M}{M-1})=t+c
\frac{M}{M-1}=e^{t}e^{c}
e^{c} is a constant we'll call C:
\frac{M}{M-1}=Ce^{t}
Solve for M=\frac{Ce^{t}}{Ce^{t}-1}
Using the IC M=1/2 when t=0:
\frac{1}{2}=\frac{Ce^{0}}{Ce^{0}-1}
Solving for C, we find C=-1
So, we have:
M=\frac{-e^{t}}{-e^{t}-1}=\frac{e^{t}}{1+e^{t}}
Does this help? Follow that?
albear
Apr 29, 2007, 07:10 AM
ln(m-1)=t+c I don't understand how you obtained the left hand side
or
M=(ce^t)/(ce^t)-1 part from the stage before it
the rest I understand thanks
galactus
Apr 29, 2007, 07:34 AM
I just integrated and solved for M.
\frac{1}{M-M^{2}}=\frac{1}{M}-\frac{1}{M-1}
Now, integrating, we get ln(M)-ln(M-1)
Which, by the properties of logs, equals:
ln(\frac{M}{M-1})
Then, I just did the e thing and solved for M. Just algebra.
albear
Apr 29, 2007, 07:55 AM
right so you split it up to get logs OK, could you explaine the
M=(ce^t)/(ce^t)-1 part and how to get to it from the stage before it please
(also I'm running into barriers with my other question ill post my working on that page if you could tell me where I've gone wrong please)
galactus
Apr 29, 2007, 08:25 AM
It's just algebra, albear.
\frac{M}{M-1}=Ce^{t}
But \frac{M}{M-1}=\frac{1}{M-1}+1
\frac{1}{M-1}+1=Ce^{t}
\frac{1}{M-1}=Ce^{t}-1
M-1=\frac{1}{Ce^{t}-1}
M=\frac{1}{Ce^{t}-1}+1=\frac{Ce^{t}}{Ce^{t}-1}
albear
Apr 29, 2007, 08:55 AM
Ahh OK thanks
galactus
Apr 29, 2007, 09:02 AM
You're welcome. I hope you learned a few littles things. Practice your algebra. It is the crux of all further math study.