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Found in: Page 271

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbit{y}}{\mathbf{-}}{\mathbf{1}}{\mathbf{,}}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{{\mathbit{e}}}^{\mathbf{x}\mathbf{+}\mathbf{y}}$

The solution is ${{\mathbit{e}}}^{{\mathbf{x}}}{\mathbf{+}}{\mathbit{y}}{{\mathbit{e}}}^{\mathbf{-}\mathbf{y}}{\mathbf{=}}{\mathbit{c}}$ .

See the step by step solution

## Step 1: Find phase plane equation

Here the system is;

$\frac{dx}{dt}=y-1\phantom{\rule{0ex}{0ex}}\frac{dy}{dt}={e}^{x+y}$

And the phase plane equation is;

$\frac{dy}{dx}=\frac{{e}^{x+y}}{y-1}$

## Step 2: Solve the equation

Here the equation is.

$\frac{dy}{dx}=\frac{{e}^{x+y}}{y-1}$

Solve by variable separating,

$\begin{array}{rcl}\int \left(y-1\right){e}^{y}dy& =& \int {e}^{x}dx\\ -y{e}^{-y}+c& =& {e}^{x}\\ {e}^{x}+y{e}^{-y}& =& c\end{array}$

This is the required result.