Answer:
a.
b. 1.5
c. 1.5
d. No
Step-by-step explanation:
a. First, let's solve the differential equation:
![\frac{dp}{dt} =3p-2p^2](https://tex.z-dn.net/?f=%5Cfrac%7Bdp%7D%7Bdt%7D%20%3D3p-2p%5E2)
Divide both sides by
and multiply both sides by dt:
![\frac{dp}{3p-2p^2}=dt](https://tex.z-dn.net/?f=%5Cfrac%7Bdp%7D%7B3p-2p%5E2%7D%3Ddt)
Integrate both sides:
![\int\ \frac{1}{3p-2p^2} dp =\int\ dt](https://tex.z-dn.net/?f=%5Cint%5C%20%5Cfrac%7B1%7D%7B3p-2p%5E2%7D%20%20dp%20%3D%5Cint%5C%20dt)
Evaluate the integrals and simplify:
![p(t)=\frac{3e^{3t} }{C_1+2e^{3t}}](https://tex.z-dn.net/?f=p%28t%29%3D%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7BC_1%2B2e%5E%7B3t%7D%7D)
Where C1 is an arbitrary constant
I sketched the direction field using a computer software. You can see it in the picture that I attached you.
b. First let's find the constant C1 for the initial condition given:
![p(0)=3=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}](https://tex.z-dn.net/?f=p%280%29%3D3%3D%5Cfrac%7B3e%5E%7B0%7D%20%7D%7BC_1%2B2e%5E%7B0%7D%20%7D%20%3D%5Cfrac%7B3%7D%7BC_1%2B2%7D)
Solving for C1:
![C_1=-1](https://tex.z-dn.net/?f=C_1%3D-1)
Now, let's evaluate the limit:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-1 } \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2-e^{-3x} }](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D-1%20%7D%20%20%5C%5C%5C%5CDivide%5Chspace%7B3%7Dthe%5Chspace%7B3%7Dnumerator%5Chspace%7B3%7Dand%5Chspace%7B3%7Ddenominator%5Chspace%7B3%7Dby%5Chspace%7B3%7De%5E%7B3t%7D%20%5C%5C%5C%5C%20%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3%20%7D%7B2-e%5E%7B-3x%7D%20%20%7D)
The expression
tends to zero as x approaches ∞ . Hence:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-1 } =\frac{3}{2} =1.5](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D-1%20%7D%20%3D%5Cfrac%7B3%7D%7B2%7D%20%3D1.5)
c. As we did before, let's find the constant C1 for the initial condition given:
![p(0)=0.8=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}](https://tex.z-dn.net/?f=p%280%29%3D0.8%3D%5Cfrac%7B3e%5E%7B0%7D%20%7D%7BC_1%2B2e%5E%7B0%7D%20%7D%20%3D%5Cfrac%7B3%7D%7BC_1%2B2%7D)
Solving for C1:
![C_1=1.75](https://tex.z-dn.net/?f=C_1%3D1.75)
Now, let's evaluate the limit:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}+1.75 } \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2+1.75e^{-3x} }](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D%2B1.75%20%7D%20%20%5C%5C%5C%5CDivide%5Chspace%7B3%7Dthe%5Chspace%7B3%7Dnumerator%5Chspace%7B3%7Dand%5Chspace%7B3%7Ddenominator%5Chspace%7B3%7Dby%5Chspace%7B3%7De%5E%7B3t%7D%20%5C%5C%5C%5C%20%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3%20%7D%7B2%2B1.75e%5E%7B-3x%7D%20%20%7D)
The expression
tends to zero as x approaches ∞ . Hence:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}+1.75 } =\frac{3}{2} =1.5](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D%2B1.75%20%7D%20%3D%5Cfrac%7B3%7D%7B2%7D%20%3D1.5)
d. To figure out that, we need to do the same procedure as we did before. So, let's find the constant C1 for the initial condition given:
![p(0)=2=\frac{3e^{0} }{C_1+2e^{0} } =\frac{3}{C_1+2}](https://tex.z-dn.net/?f=p%280%29%3D2%3D%5Cfrac%7B3e%5E%7B0%7D%20%7D%7BC_1%2B2e%5E%7B0%7D%20%7D%20%3D%5Cfrac%7B3%7D%7BC_1%2B2%7D)
Solving for C1:
![C_1=-\frac{1}{2} =-0.5](https://tex.z-dn.net/?f=C_1%3D-%5Cfrac%7B1%7D%7B2%7D%20%3D-0.5)
Can a population of 2000 ever decline to 800? well, let's find the limit of the function when it approaches to ∞:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-0.5 } \\\\Divide\hspace{3}the\hspace{3}numerator\hspace{3}and\hspace{3}denominator\hspace{3}by\hspace{3}e^{3t} \\\\ \lim_{t \to \infty} \frac{3 }{2-0.5e^{-3x} }](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D-0.5%20%7D%20%20%5C%5C%5C%5CDivide%5Chspace%7B3%7Dthe%5Chspace%7B3%7Dnumerator%5Chspace%7B3%7Dand%5Chspace%7B3%7Ddenominator%5Chspace%7B3%7Dby%5Chspace%7B3%7De%5E%7B3t%7D%20%5C%5C%5C%5C%20%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3%20%7D%7B2-0.5e%5E%7B-3x%7D%20%20%7D)
The expression
tends to zero as x approaches ∞ . Hence:
![\lim_{t \to \infty} \frac{3e^{3t} }{2e^{3t}-0.5 } =\frac{3}{2} =1.5](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B3e%5E%7B3t%7D%20%7D%7B2e%5E%7B3t%7D-0.5%20%7D%20%3D%5Cfrac%7B3%7D%7B2%7D%20%3D1.5)
Therefore, a population of 2000 never will decline to 800.