Q27E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Using the software, sketch the direction field in the phase-plane for the system $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{5}\mathit{x}\mathbf{-}\mathbf{4}\mathit{y}$. From the sketch, predict the asymptotic limit (as $\mathit{t}\mathbf{\to }\mathbf{\infty }$

 of the solution starting at (1, 1).

Step-by-Step Solution

Answer

The solution starts at (1,1) and moves towards zero as $\mathbf{t}\to \infty$ the solution approaches to (0,0).

Step 1: Find the critical point

Here the equation is:

 $$\begin{gathered} \frac{dx}{dt}=-2x+y \\ \frac{dy}{dt}=-5x-4y \end{gathered}$$ 

And 

 $\frac{dy}{dx}=\frac{-5x-4y}{-2x+y}$

 

The solution starts at (1,1) and moves towards zero as the solution approaches to (0,0).

Step 2: Sketch the Directional field

This is the required result.