Chapter 6: Q. 21 (page 575)

Solve the initial-value problem
$\frac{d y}{d x} = 2 y (1 - 5 y), y (0) = 1$

Short Answer

Expert verified

The solution is $y (x) = \frac{e^{2 x}}{5 e^{2 x} - 4}$

Step by step solution

Step 1. Given information

given value is $\frac{d y}{d x} = 2 y (1 - 5 y)$

Step 2. Solve equation

$\int \frac{d y}{y (1 - 5 y)} = 2 \int d x \frac{1}{y (1 - 5 y)} = \frac{A}{y} + \frac{B}{1 - 5 y} A = 1 B = 5 \frac{1}{y (1 - 5 y)} = \frac{1}{y} + \frac{5}{1 - 5 y} \int [\frac{1}{y} + \frac{5}{1 - 5 y}] d y = 2 x + c \ln (y) - \ln (|1 - 5 y|) = 2 x + c \ln (|\frac{y}{1 - 5 y}|) = 2 x + c \frac{y}{1 - 5 y} = e^{2 x + c} \frac{1 - 5 y}{y} = e^{- 2 x - c} = A e^{- 2 x} \frac{1}{y} = 5 + A e^{- 2 x} y = \frac{1}{5 + A e^{- 2 x}} 1 = \frac{1}{5 + A} 5 + A = 1 A = - 4 y (x) = \frac{1}{5 - 4 e^{- 2 x}} y (x) = \frac{e^{2 x}}{5 e^{2 x} - 4}$

Step 3. The solution

The solution is $y (x) = \frac{e^{2 x}}{5 e^{2 x} - 4}$

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Solve the initial-value problem
$\frac{d y}{d x} = 2 y (1 - 5 y), y (0) = 1$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve equation

Step 3. The solution

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Solve the initial-value problemdydx=2y1-5y,y0=1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve equation

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

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Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Solve the initial-value problem
$\frac{d y}{d x} = 2 y (1 - 5 y), y (0) = 1$