Q9E - 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. $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{2}\mathit{y}\mathbf{-}\mathit{x}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathbf{+}\mathit{y}$

Step-by-Step Solution

Answer

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

Step 1: Find phase plane equation

Here the system is;

 $$\begin{gathered} \frac{dx}{dt}=2y-x \\ \frac{dy}{dt}=e^x+y \end{gathered}$$

And the phase plane equation is;

$\frac{dy}{dx}=\frac{e^x+y}{2y-x}$ 

Step 2: Solve the equation

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

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

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 $e^x-xy-y^2+c$.