Question

Let the population of rabbits surviving at a time t be governed by the differential equation `(dp(t))/dt = 1/2p(t) - 200`. If p(0) = 100, then p(t) equals ______ 

विकल्प

• `600 - 500e^{t/2}`

• `400 - 300e^{(-t)/2}`

• `400 - 300e^{t/2}`

• `300 - 200e^{(-t)/2}`

Answer 1

Let the population of rabbits surviving at a time t be governed by the differential equation `(dp(t))/dt = 1/2p(t) - 200`. If p(0) = 100, then p(t) equals `underline(400 - 300e^{t/2})`.

Explanation:

`(dp(t))/dt = 1/2p(t) - 200`

Integrating on both sides, we get

`int(d(p(t)))/(1/2p(t) - 200) = intdt + c_1`

⇒ `2log((p(t))/2 - 200) = t + c_1`

= `(p(t))/2 - 200 = e^{t/2} c, ("where"  c = e^{(c_1)/2})` ................(i)

Putting t = 0, we get

`(p(0))/2 - 200 = e^0 c`

⇒ `100/2 - 200 = c ⇒ c = -150`

∴ `(p(t))/2 - 200 = e^{1/2}(-150)` ........[From (i)]

⇒ p(t) = `400 - 300e^{t/2}`