Q26E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Using the software, sketch the direction field in the phase-plane for the system $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{-}{\mathit{x}}^{\mathbf{3}}\mathbf{.}$ From the sketch, conjecture whether all solutions of this system are bounded. Solve the related phase plane differential equation and confirm your conjecture.

Step-by-Step Solution

Answer

The solutions of the system are bounded.

Step 1: Find a critical point

 

Here the equation is:

 $$\begin{gathered} \frac{dx}{dt}=y \\ \frac{dy}{dt}=-x-x^3 \end{gathered}$$ 

And

$$\begin{gathered} \frac{dy}{dx}=\frac{-x-x^3}{y} \\ \int ydy=\int -x-x^{3}dx \\ \frac{y^2}{2}=-\frac{x^2}{2}-\frac{x^4}{4}+c \end{gathered}$$
 

Step 2: Sketch the Directional field.

Step 3: Sketch for the solution.

Thus, the system is bounded.