The population p(t) at time t of a certain mouse species satisfies the differential equation \frac{dp\left ( t \right )}{dt}= 0.5\; p(t)-450.

If p(0) = 850, then the time at which the population becomes zero is

  • Option 1)

    2ln18

  • Option 2)

    ln9

  • Option 3)

     \frac{1}{2}ln18

  • Option 4)

    ln18

 

Answers (1)

As we learnt in 

Solution of Differential Equation -

\frac{\mathrm{d}y }{\mathrm{d} x} =f\left ( ax+by+c \right )

put

 Z =ax+by+c

 

 

- wherein

Equation with convert to

\int \frac{dz}{bf\left ( z \right )+a} =x+c

 

 

 

 

given \frac{d\left ( p\left ( t \right ) \right )}{dt}= 0.5p\left ( t \right )-450

 

\therefore \int_{850}^{p}\frac{2dp}{p-900}= \int_{0}^{t}dt\Rightarrow 2ln\frac{p-900}{-50}=t

\Rightarrow p= 900-50.e^{t/2}

I.F.\ p=0,\ then \frac{900}{50}= e^{t/2}\Rightarrow t= 2\ ln18


Option 1)

2ln18

This option is correct.

Option 2)

ln9

This option is incorrect.

Option 3)

 \frac{1}{2}ln18

This option is incorrect.

Option 4)

ln18

This option is incorrect.

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