Question:

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

Updated On: Jul 6, 2022
  • $ 600 - 500 \,e^{t/2}$
  • $400 - 300\, e^{-t/2} $
  • $400 - 300 \, e^{t/2 }$
  • $300 - 200 \, e^{-t/2.} $
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The Correct Option is C

Solution and Explanation

Since $\frac{dp}{dt} - \frac{1}{2} p\left(t\right) = -200$ is lmear in $y$ $\therefore I.F. = e^{\int \frac{1}{2} dt} = e^{-\frac{t}{2}}$ $\therefore$ role is $p\cdot e^{\frac{-t}{2}} = \int -200\cdot\left(e^{-\frac{t}{2}}\right)dt + C$ $= -200\cdot \frac{e^{-t/ 2}}{-1/ 2}+C$ $= 400 \,e^{-t/2} + C$ Since $p\left(0\right) = 100$ $\therefore 100 \,e^{0} = 400\, e^{0} + C$ $\Rightarrow 100 = 400 + C$ $\Rightarrow C = - 300$ $\therefore p\left(t\right) = 400 - 300 \,e^{t/2}$
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Notes on Differential equations

Concepts Used:

Differential Equations

A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.

Orders of a Differential Equation

First Order Differential Equation

The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’

Second-Order Differential Equation

The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.

Types of Differential Equations

Differential equations can be divided into several types namely

  • Ordinary Differential Equations
  • Partial Differential Equations
  • Linear Differential Equations
  • Nonlinear differential equations
  • Homogeneous Differential Equations
  • Nonhomogeneous Differential Equations