Question

\(\frac{d P}{d t}=0.0008 P(700-P) \text { and } P=10 \text { when } t=0\)

Short answer

The solution to the differential equation is \( P(t) = \frac{700 \cdot 10}{10+690 \cdot e^{-0.0008t}} \)

Step-by-step solution

Separate the Variables

Rewrite the differential equation in a form where all terms involving P are on one side and all terms involving t are on the other. This gives: \( \frac{d P}{P(700-P)}=0.0008 dt \)

Integrate Both Sides

Now that the variables are separated, integrate both sides of the equation. This results in: \( \int{\frac{1}{P(700-P)} dP} = \int{0.0008 dt} \) or \( \ln(\frac{P}{700-P})=-0.0008t+ C \)

Use the Initial Condition to find C

The constant of integration C can be found by substituting the initial condition into the integrated equation. Given P=10 when t=0, we get: \( \ln(\frac{10}{700-10}) = C \)

Substitute C into the equation

Replace the C in the equation found in Step 2 with the value from Step 3 to get the particular solution: \( \ln(\frac{P}{700-P})=-0.0008t+ \ln(\frac{10}{700-10}) \)

Simplify the equation

By applying the properties of logarithms and simplifying, the express the P as a function of t, rearrange to get: \( P(t) = \frac{700 \cdot 10}{10+690 \cdot e^{-0.0008t}} \)