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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q 17RP (page 79)

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

Short Answer

Expert verified

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

Step by step solution

01

Given information and simplification

Given that, $\frac{dy}{d 胃} + 2 y = y^{2} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$ $\frac{dy}{d 胃} + 2 y = y^{2} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

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

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

$y^{- 2} \frac{dy}{d 胃} + 2 y^{- 1} = 1 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

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

$\begin{matrix} - \frac{du}{d 胃} + 2 u = 1 \\ \frac{du}{d 胃} - 2 u = - 1 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (3) \end{matrix}$

Let $P (胃) = - 2$

Find the value of $渭 (胃)$ .

$\begin{matrix} 渭 (胃) = e^{鈭� P (胃) d 胃} \\ = e^{- 鈭� 2 d 胃} \\ = e^{- 2 胃} \end{matrix}$

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

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

02

integration method

Now integrate the equation on both sides.

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

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

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