Q. 76 聽Prove Theorem 6.21 by solving ... [FREE SOLUTION]

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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

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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

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

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

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

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91影视

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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91影视

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

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Applied Mathematics

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

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