SATHEE: Differential Equations 3 Question 3

Differential Equations 3 Question 3

3. Let the population of rabbits surviving at a time $\frac{d p (t)}{d x} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ is equal to

(a) $400 - 300 e^{\frac{1}{2}}$

(b) $300 - 200 e^{- \frac{t}{2}}$

(d) $400 - 300 e^{- \frac{t}{2}}$

(2014 Main)

Show Answer

Answer:

Correct Answer: 3. (a)

Solution:

Given, differential equation is $\frac{d p}{d t} - \frac{1}{2} p (t) = - 200$ is a linear differential equation.

Here,

$\begin{aligned} p (t) & = \frac{- 1}{2}, Q (t) = - 200 \\ I F & = e^{\int - \frac{1}{2} d t} = e^{- \frac{t}{2}} \end{aligned}$

Hence, solution is

$\begin{aligned} p (t) \cdot I F & = \int Q (t) \cdot I F d t \\ p (t) \cdot e^{- \frac{t}{2}} & = \int - 200 \cdot e^{- \frac{t}{2}} d t \\ p (t) \cdot e^{- \frac{t}{2}} & = 400 e^{- \frac{t}{2}} + K \\ p (t) & = 400 + k e^{- 1 / 2} \\ If p (0) = 100, then k & = - 300 \\ \Rightarrow p (t) & = 400 - 300 e^{\frac{t}{2}} \end{aligned}$

Differential Equations 3 Question 3

3. Let the population of rabbits surviving at a time $\frac{d p (t)}{d x} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Differential Equations 3 Question 3

3. Let the population of rabbits surviving at a time dp(t)dx=12p(t)−200. If p(0)=100, then p(t) is equal to

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

3. Let the population of rabbits surviving at a time $\frac{d p (t)}{d x} = \frac{1}{2} p (t) - 200$ . If $p (0) = 100$ , then $p (t)$ is equal to