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Q 17RP

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 79

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Question: In Problems 1-30, solve the equation.
$\frac{dy}{d θ} + 2 y = y^{2}$

The solution of the given equation is $y = \frac{2}{1 + {Ce}^{2 θ}}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information and simplification

Given that, $\frac{dy}{d θ} + 2 y = y^{2} \cdot \cdot \cdot \cdot \cdot \cdot (1)$ $\frac{dy}{d θ} + 2 y = y^{2} \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Since, the given equation is the Bernoulli equation with n = 2, $P (θ) = 2$ and $Q (θ) = 1$ .

Now divide y² on both sides of the equation (1).

$y^{- 2} \frac{dy}{d θ} + 2 y^{- 1} = 1 \cdot \cdot \cdot \cdot \cdot \cdot (2)$

Let us take $u = y^{- 1}$ and $- \frac{du}{d θ} = y^{- 2} \frac{dy}{d θ}$ . Then,

$\begin{matrix} - \frac{du}{d θ} + 2 u = 1 \\ \frac{du}{d θ} - 2 u = - 1 \cdot \cdot \cdot \cdot \cdot \cdot (3) \end{matrix}$

Let $P (θ) = - 2$

Find the value of $μ (θ)$ .

$\begin{matrix} μ (θ) = e^{\int P (θ) d θ} \\ = e^{- \int 2 d θ} \\ = e^{- 2 θ} \end{matrix}$

Multiply $e^{- 2 θ}$ in equation (3) on both sides.

$\begin{matrix} e^{- 2 θ} \frac{du}{d θ} - 2 e^{- 2 θ} u = - e^{- 2 θ} \\ \frac{du}{d θ} [e^{- 2 θ} u] = - e^{- 2 θ} \end{matrix}$

Step 2: integration method

Now integrate the equation on both sides.

$\begin{matrix} \int \frac{du}{d θ} [e^{- 2 �} u] d θ = - \int e^{- 2 θ} d θ \\ e^{- 2 θ} u = \frac{e^{2 θ}}{2} + C_{1} \\ u = \frac{1}{2} + C_{1} e^{2 θ} \\ y^{- 1} = \frac{1 + {Ce}^{2 θ}}{2} \\ y = \frac{2}{1 + {Ce}^{2 θ}} \end{matrix}$

So, the solution is $y = \frac{2}{1 + {Ce}^{2 θ}}$

Most popular questions for Math Textbooks

Use the method discussed under “Homogeneous Equations” to solve problems 9-16.

$\frac{dy}{dx} = \frac{y (lny - lnx + 1)}{x}$

Question: In Problems 1 - 30, solve the equation.

$\frac{dx}{dt} = 1 + \cos^{2} (t - x)$

Question: In problems 33-40 Solve the equation given in Problem 3.

$\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$

Use the method discussed under “Homogeneous Equations” to solve problems 9-16 $(xy + y^{2}) dx - x^{2} dy = 0$

Question: In Problems 33–40, solve the equation given in: Problem 7.

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