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

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system. $\frac{d x}{d t} = 2 y - x, \frac{d y}{d t} = e^{x} + y$

The solution is $e^{x} - x y - y^{2} + c$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find phase plane equation

Here the system is;

$\frac{d x}{d t} = 2 y - x \frac{d y}{d t} = e^{x} + y$

And the phase plane equation is;

$\frac{d y}{d x} = \frac{e^{x} + y}{2 y - x}$

Step 2: Solve the equation

Here the equation is $\frac{d y}{d x} = \frac{e^{x} + y}{2 y - x}$ .

role="math" localid="1663962144511" $\begin{array}{rcl} (e^{x} + y) d x + (x - 2 y) d y & = & 0 \\ M & = & (e^{x} + y) \\ N & = & (x - 2 y) \\ \frac{\partial M}{\partial y} & = & 1 = \frac{\partial N}{\partial x} \end{array}$

The equations are exact.

Step 3: Find the value of F and G.

Now,

$\begin{array}{rcl} F (x, y) & = & \int M (x, y) d x + g (y) \\ = & \int (e^{x} + y) d x + g (y) \\ = & e^{x} + x y + g (y) \\ N (x, y) & = & x + g' (y) \\ (x - 2 y) & = & x + g' (y) \\ g' (y) & = & - 2 y \\ g (y) & = & - y^{- 2} + c \\ F (x, y) & = & e^{x} - x y - y^{2} + c \end{array}$

Therefore, the solution is $e^{x} - x y - y^{2} + c$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system. $\frac{d x}{d t} = 2 y - x, \frac{d y}{d t} = e^{x} + y$

Step by Step Solution

Step 1: Find phase plane equation

Step 2: Solve the equation

Step 3: Find the value of F and G.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system. dxdt=2y-x,dydt=ex+y

Step by Step Solution

Step 1: Find phase plane equation

Step 2: Solve the equation

Step 3: Find the value of F and G.

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In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system. $\frac{d x}{d t} = 2 y - x, \frac{d y}{d t} = e^{x} + y$