Solve differential equation explicitly: dP/dt = 0.2P (1 - P/1000) - 48, using P_0 = 350

[Elix]

I honestly don't know how I keep getting huge numbers.
Equation:

Problem:

Here's all the work I've done:

 

[MarkFL]

My old eyes have a hard time with your image.

We are given to solve the IVP:

[MATH]\d{P}{t}=0.2P\left(1-\frac{P}{1000}\right)-48[/MATH] where \(P(0)=P_0\)

Let's write the ODE as:

[MATH]\d{P}{t}=-\frac{(P-400)(P-600)}{5000}[/MATH]
Thus, separating variables and switching dummy variables, and integrating on the given boundaries:

[MATH]5000\int_{P_0}^{P}\frac{1}{(P-400)(P-600)}\,du=-\int_0^t\,dv[/MATH]
[MATH]\ln\left(\frac{(P-600)(P_0-400)}{(P-400)(P_0-600)}\right)=-\frac{2}{50}t[/MATH]
[MATH]\frac{(P-600)(P_0-400)}{(P-400)(P_0-600)}=e^{-\frac{2}{50}t}[/MATH]
[MATH]\frac{P-600}{P-400}=\frac{P_0-600}{P_0-400}e^{-\frac{2}{50}t}[/MATH]
[MATH]P-600=\frac{P_0-600}{P_0-400}e^{-\frac{2}{50}t}(P-400)[/MATH]
[MATH]P\left(1-\frac{P_0-600}{P_0-400}e^{-\frac{2}{50}t}\right)=600-400\frac{P_0-600}{P_0-400}e^{-\frac{2}{50}t}[/MATH]
[MATH]P\left((P_0-400)-(P_0-600)e^{-\frac{2}{50}t}\right)=600(P_0-400)-400(P_0-600)e^{-\frac{2}{50}t}[/MATH]
[MATH]P(t)=\frac{600(P_0-400)-400(P_0-600)e^{-\frac{2}{50}t}}{(P_0-400)-(P_0-600)e^{-\frac{2}{50}t}}[/MATH]
[MATH]P(t)=\frac{600(P_0-400)e^{\frac{2}{50}t}-400(P_0-600)}{(P_0-400)e^{\frac{2}{50}t}-(P_0-600)}[/MATH]
Now, you can just plug in for \(P_0\).

 

[Dr.Peterson]

I think you took too large a step in getting to [MATH]\left|\frac{P-600}{P-400}\right| = \frac{1}{200}e^{-\frac{1}{5000}t + C}[/MATH]. Try writing an extra step or two and see what you get.