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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. $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{2}}{\mathbit{y}}{\mathbf{-}}{\mathbit{x}}{\mathbf{,}}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{{\mathbit{e}}}^{{\mathbf{x}}}{\mathbf{+}}{\mathbit{y}}$

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

See the step by step solution

## Step 1: Find phase plane equation

Here the system is;

$\frac{dx}{dt}=2y-x\phantom{\rule{0ex}{0ex}}\frac{dy}{dt}={e}^{x}+y$

And the phase plane equation is;

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

## Step 2: Solve the equation

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

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

The equations are exact.

## Step 3: Find the value of F and G.

Now,

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

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