Question

The population ( p ) of the city at time ( t ) is given by ( \frac{dp}{dt} = \frac{p}{2} - 100 ). If initial population is 100 then ( p = )

• A. ( 200 + 100e^{t/2} )
• B. ( 200 - 100e^{t/2} )
• C. ( 300 - 100e^{t/2} )
• D. ( 300 + 100e^{t/2} )

Hint:

Differential equations of the form $\frac{dy}{dx} = ay + b$ always have exponential solutions.

Answer 1

The Correct Option is B

Step 1: Separate variables
(\frac{dp}{p - 200} = \frac{dt}{2}).

Step 2: Integrate
(\ln|p - 200| = \frac{t}{2} + C).
(p - 200 = Ae^{t/2}).

Step 3: Use initial condition
At (t = 0, p = 100):
(100 - 200 = A e^0 \implies A = -100).

Step 4: Formulate final equation
(p = 200 - 100e^{t/2}).
Final Answer: (B)