Q7E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.

$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{-}\mathbf{1}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}{\mathit{e}}^{\mathbf{x}\mathbf{+}\mathbf{y}}$

Step-by-Step Solution

Answer

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

Step 1: Find phase plane equation

Here the system is;

$$\begin{gathered} \frac{dx}{dt}=y-1 \\ \frac{dy}{dt}=e^{x+y} \end{gathered}$$ 

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,

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

This is the required result.