1.

Let thepopulation of rabbits surviving at a time t be governed by the differentialequation `(d p(t)/(dt)=1/2p(t)-200.`If `p(0)""=""100`, then p(t) equals(A. `600-500e^(t//2)`B. `400-300e^(-t//2)`C. `400-300e^(t//2)`D. `300-200e^(-t//2)`

Answer» Correct Answer - C
We have,
`(d)/(dt)(f(t))=(1)/(2)p(t)-200`
`rArr" "(d)/(dt)(p(t))+(-(1)/(2))p(t)=-200" …(i)"`
This is a linear differential equation with I.F. `=e^(int-(1)/(2)dt)=e^(-(1)/(2))`
Multiplying both sides of (i) by I.F. `=e^(-t//2)`, we obtain
`e^(-t//2)(d)/(dt)(p(t))+(-(1)/(2))p(t)e^(-t//2)=-200e^(-t//2)`
Integrating both sides with respect to t, we get
`P(t)e^(-t//2)=400e^(-t//2)+C" ...(ii)"`
Putting t = 0 and p(0) = 100, we get
`100=400+CrArr C=-300`
Putting C =` -300`, we get
`p(t)e^(-t//2)=400e^(-t//2)-300`
`rArr" "p(t)=400-300e^(t//2)`


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