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(A) The rate of change of population $p$ with respect to time $t$ is proportional to the population itself,given by the differential equation: $\frac{

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$p = C e^{\frac{3t}{100}}$

Vedclass · Answer

(A) The rate of change of population $p$ with respect to time $t$ is proportional to the population itself,given by the differential equation: $\frac{dp}{dt} = \frac{3}{100} p$. Separating the variables,we get $\frac{dp}{p} = \frac{3}{100} dt$. Integrating both sides,we have $\int \frac{dp}{p} = \int \frac{3}{100} dt$. This results in $\ln(p) = \frac{3t}{100} + K$,where $K$ is the constant of integration. Exponentiating both sides,we get $p = e^{\frac{3t}{100} + K} = e^K \cdot e^{\frac{3t}{100}}$. Letting $C = e^K$,we obtain the equation $p = C e^{\frac{3t}{100}}$.

The population of a city increases at the rate of $3 \%$ per year. If the population is $p$ at time $t$,then the equation of $p$ in terms of $t$ is . . . . . . .

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