Solve this model to obtain an equation as a function of time

[Paupe1]

The problemart 1
dE/dt= B(A-E) Where A and B are constants, solve this model to obtain an equation for E as a function of time. Use the initial condition that, at t= 0, E= E⁰.

My answerart 1
E(t)= A+Ce^-Bt If I use E⁰=1 at t= 0, then
(1)=A+Ce^-B(0) Cancelling out e
1= A+C Which leaves me with
C= 1-A And the equation as a function of time would be

E(t)= A+(1-A)e^-Bt Am I on the right track? The reason I ask is that part 2 messes everything up for me, and it's making me question whether or not I'm even doing part 1 correctly. Here is part 2.

The problemart 2
Convert the equation from part 1 to provide an equation for E_T as a function of time given the relationship between the two as E=fE_T, where f=2/3. I am also given a data table with values

t	0	1.5	2.5	3.5	4.5
ET	0.026	0.023	0.019	0.016	0.012

My Answer: Part 2
(2/3)E_T(t)= A+(1-A)e^-Bt Substituting (2/3)E_T for E, and plugging in t=0 and its corresponding E_T=0.026
(2/3)(0.026)= A+(1-A)e^-B(0) And I end up with
0=0.98, Which is obviously wrong. If I can get the first part equation and second part equation set up correctly as functions of time, having part 2 changed with the given relationship, I can solve for A and B, but setting up those two functions are really messing with me. Here is a link for some help I've been able to get, thanks for any help that is available. http://www.math.montana.edu/frankw/ccp/calculus/des/genpart/learn.htm (EXAMPLE 3)

 

[HallsofIvy]

The problemart 1
dE/dt= B(A-E) Where A and B are constants, solve this model to obtain an equation for E as a function of time. Use the initial condition that, at t= 0, E= E⁰.

My answerart 1
E(t)= A+Ce^-Bt

HOW did you get that?

If I use E⁰=1 at t= 0, then
(1)=A+Ce^-B(0) Cancelling out e
1= A+C Which leaves me with
C= 1-A And the equation as a function of time would be

E(t)= A+(1-A)e^-Bt Am I on the right track? The reason I ask is that part 2 messes everything up for me, and it's making me question whether or not I'm even doing part 1 correctly. Here is part 2.

The problemart 2
Convert the equation from part 1 to provide an equation for E_T as a function of time given the relationship between the two as E=fE_T, where f=2/3. I am also given a data table with values

t		1.5	2.5	3.5	4.5
ET	0.026	0.023	0.019	0.016	0.012

My Answer: Part 2
(2/3)E_T(t)= A+(1-A)e^-Bt Substituting (2/3)E_T for E, and plugging in t=0 and its corresponding E_T=0.026
(2/3)(0.026)= A+(1-A)e^-B(0) And I end up with
0=0.98, Which is obviously wrong. If I can get the first part equation and second part equation set up correctly as functions of time, having part 2 changed with the given relationship, I can solve for A and B, but setting up those two functions are really messing with me. Here is a link for some help I've been able to get, thanks for any help that is available. http://www.math.montana.edu/frankw/ccp/calculus/des/genpart/learn.htm (EXAMPLE 3)

So \(\displaystyle E_\tau(t)= \frac{3}{2}E(t)\)? But then \(\displaystyle E_\tau(0)= \frac{3}{2}E(0)\) which is clearly NOT .026.

Perhaps you are intended to go back to \(\displaystyle E= A+ Ce^{-Bt}\) and find new values of A, B, and C to fit.

 

[Paupe1]

The problemart 1
dE/dt= B(A-E) Where A and B are constants, solve this model to obtain an equation for E as a function of time. Use the initial condition that, at t= 0, E= E⁰.

My answerart 1
E(t)= A+Ce^-Bt

How I got this...
Using this website for help, http://www.math.montana.edu/frankw/ccp/calculus/des/genpart/learn.htm , Example 1 and 3, I somewhat tried to mimic what was done there to set up the first equation as a function of time, which led me to the answer I wrote. But on the second part, it wants me to convert the 1st equation into a second equation using the relationship of E=fE_T, where f=2/3. Since my table only gives values for t in relation to E_T, I'm stuck on trying to figure out how to find any values for the 1st equation, unless the 1st equation is just there to further change into a second equation using E_T. The problem I'm having is just with setting up the first 2 equations. I'm wondering if it is just as simple as just taking the equation I got for the first part, and just substituting E with (2/3)E_T. I really appreciate the help if you can help or not.

 

[Deleted member 4993]

dE/dt= B(A-E) Where A and B are constants, solve this model to obtain an equation for E as a function of time. Use the initial condition that, at t= 0, E= E⁰.

dE/dt = B(A-E)

dE/(A-E) = B dt

ln[C₁*(A-E)] = -Bt

A - E = C * e^-Bt

Applying initial condition → C = A - E₀

A - E = (A-E₀) * e^-Bt

E = A - (A-E₀) * e^-Bt

Now continue....

 

[Paupe1]

dE/dt = B(A-E)

dE/(A-E) = B dt

ln[C₁*(A-E)] = -Bt

A - E = C * e^-Bt

Applying initial condition → C = A - E₀

A - E = (A-E₀) * e^-Bt

E = A - (A-E₀) * e^-Bt

Now continue....

Ok, I was able to work out how you got the first equation. I've spent a few days attempting to make sense of the second part, and the best fit I could work out is the following:

Since E = (2/3)E_T, and it tells me to remember to convert E₀ to E_T⁰, then the 2nd equation should be...
(2/3)E_T = A - (A-E_T⁰) * e^-Bt

Given the data table, and the problem telling me to use the value for E_T⁰ at t=0 of that table, then...
(2/3)E_T = A - [A-(0.026)] * e^-B(0) And I solved for E_T, which came to E_T = 0.039 and I thought I was okay... but then, to solve for A & B, and also being told that at any instant in time E is in equilibrium, I end up with a negative natural log, which is undefined... here is what I tried using the table at t=1.5, thinking that E_T on the left part of the equation remains constant, solving for B.

(2/3)(0.039) = A - [A-(0.023)] * e^-B(1.5)
(A-0.026)/(A-0.023) = e^-B(1.5)
ln(A-0.026) - ln(A-0.023) = -B(1.5)
A - 0.026 - A + 0.023 = e^-B(1.5)
-0.003 = e^-B(1.5)

I'm wondering if somewhere I'm supposed to take an absolute value, turning the -0.026 into a positive value. Thanks in advance for any help, I hope I provided correct and complete information on the problem.