Final answer:
To solve the given differential equation, we start by separating the variables and integrating both sides. Then, we isolate P and apply exponential operations to the equation. Finally, substituting the given values into the equation, we obtain the specific solution P(t) = 600e^{-0.25t}.
Step-by-step explanation:
To solve the differential equation dP/dt = c ln(K/P) P, we can start by separating the variables. Moving all the P terms to one side and the t terms to the other side, we have dP/(ln(K/P) P) = c dt. Next, we can integrate both sides of the equation.
On the left side, we can use the substitution u = ln(K/P) so that du = -1/(P ln(10)) dP. Integrating both sides, we get ∫-1/(P ln(10)) dP = ∫c dt, which simplifies to -ln(10) ln(K/P) = ct + C.
Now, we can solve for P by isolating it. Taking the exponential function of both sides, e^{-ln(10) ln(K/P)} = e^{ct + C}. Simplifying further, we have (K/P)^{ln(10)} = e^{ct+C}, which becomes K/P = e^{ct+C}. Rearranging this equation, we get P/K = e^{-(ct+C)}, and isolating P, we have P = Ke^{-(ct+C)} = Ke^{-ct}e^{-C}.
Since e^{-C} is another constant, we can rewrite P = Ke^{-ct}e^{-C} as P = A e^{-ct}, where A = Ke^{-C}. Given that c = 0.25, K = 4000, and P_0 = 600, we can substitute these values into the equation to get the specific solution. The final equation is P(t) = 600e^{-0.25t}.