Chapter 6: Q. 76 (page 572)

Prove Theorem 6.21 by solving the initial-value problem $\frac{d P}{d t} = r P 1 - \frac{P}{K}$ with P(0) = P0, where r and K are constants

Short Answer

Expert verified

Proved

Step by step solution

Step 1. Given

The given initial-value problem is $\frac{d P}{d t} = r P 1 - \frac{P}{K}$ .

Step 2. Proof

Observe that the differential equation does not involve the independent variable at all, so solve the differential equation by antidifferentiation method

$\int \frac{d P}{P (1 - \frac{P}{K})} = r d t$

The Integrand on the left hand side needs to be simplified by the use of partial fractions. So, first resolve the fraction in to partial fractions by using cover up rule

$\frac{1}{P (1 - \frac{P}{K})} = \frac{A}{P} + \frac{B}{1 - \frac{P}{K}} A = 1 B = \frac{1}{K} H ence, Integral on the left hand side comprises of two integrals . Solve these integrals as \int \frac{1}{P} dP + \frac{1}{K} \int \frac{dP}{1 - \frac{P}{K}} = r \int dt \ln |p| + k \ln |1 - \frac{P}{K}| = rt + C_{1} simplifying the equations P = \frac{K}{1 + {Ae}^{- rt}} Now take t = 0, P = P_{0} in the above result and evaluate the constant A P_{0} = \frac{K}{1 + A} A = \frac{K - P_{0}}{P_{0}} P (t) = \frac{{KP}_{0}}{P_{0} + (K - P_{0}) e^{- rt}}$

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Prove Theorem 6.21 by solving the initial-value problem $\frac{d P}{d t} = r P 1 - \frac{P}{K}$ with P(0) = P0, where r and K are constants

Short Answer

Step by step solution

Step 1. Given

Step 2. Proof

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Prove Theorem 6.21 by solving the initial-value problem dPdt=rP1-PKwith P(0) = P0, where r and K are constants

Short Answer

Step by step solution

Step 1. Given

Step 2. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Prove Theorem 6.21 by solving the initial-value problem $\frac{d P}{d t} = r P 1 - \frac{P}{K}$ with P(0) = P0, where r and K are constants