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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

01

Step 1. Given information

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

02

Step 2. Solve equation

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

03

Step 3. The solution

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

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Most popular questions from this chapter

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