Question

In Exercises \(1-4\) find the equations of the trajectories of the given undamped equation. Identify the equilibrium solutions, determine whether they are stable or unstable, and plot some trajectories. $$ \mathrm{C} / \mathrm{G} \quad y^{\prime \prime}+y^{3}=0 $$

Short answer

In summary, we found the equilibrium solution for the equation \(y^{\prime\prime} + y^3 = 0\) to be y=0. However, we could not determine the stability of the solution. To visualize the trajectories, rewrite the given equation in terms of y'(t) and plot it on a phase plane using software or online tools.

Step-by-step solution

Identify the equilibrium solutions

To find equilibrium solutions, we have to find the values of y for which the given equation \(y^{\prime\prime} + y^3 = 0\) holds true when \(y^{\prime\prime}=0\). Setting \(y^{\prime\prime}\) to 0, we get: \(y^3=0\) This implies y = 0. Hence, the equilibrium solution is y = 0.

Determine the stability of the equilibrium solution

To analyze the stability of the equilibrium solution, we'll look at the second derivative of y. If the second derivative is positive, it indicates that the solution is stable, and if the second derivative is negative, the solution is unstable. Recall the given equation: $$y^{\prime\prime} + y^3 = 0$$ Now, take the first derivative of both sides with respect to y: $$\frac{d}{dy}(y^{\prime\prime} + y^{3}) = \frac{d}{dy}(0)$$ We obtain: $$y^{\prime\prime\prime} + 3y^2\frac{dy}{dt} = 0$$ Let y=0, then: $$y^{\prime\prime\prime}(0) = 0$$ As the second derivative of y at the equilibrium solution is equal to 0, we cannot determine if the equilibrium solution is stable or unstable in this case.

Plot some trajectories

To plot the trajectories, we'll use a phase plane, with y on the vertical axis and y'(t) on the horizontal axis. Since the given equation is in terms of y(t), we need to rewrite it in terms of y'(t): Recall the given equation: $$y^{\prime\prime} + y^3 = 0$$ Let $$v = y'$$, then $$v'=y^{\prime\prime}$$. The equation then becomes: $$v' + y^3 = 0$$ Now we have the equation in the form v'(y) and can plot the trajectories on the phase plane. To plot the trajectories, one can use some software or online tools, like Desmos, Python or any other preferred plotting tool.

Key concepts

Equilibrium Solutions

When dealing with differential equations, equilibrium solutions are central points where the system remains constant over time. For our equation, \(y'' + y^3 = 0\), we aim to find values of \(y\) where the derivatives do not change. Essentially, we want \(y'' = 0\) and \(y^3 = 0\). By substituting \(y'' = 0\) into the equation, we simplify to \(y^3 = 0\), leading to \(y = 0\).

This indicates that the system is in equilibrium when \(y = 0\). Understanding equilibrium solutions is vital as they serve as reference points for analyzing the stability and behavior of the system's trajectories around these points.

Stability Analysis

Stability analysis examines how a system behaves when slightly perturbed from its equilibrium solution. We want to know if our system returns to equilibrium, stays away, or behaves indifferently. In our exercise, the second derivative \(y'' + y^3 = 0\) leads to \(y'' = -y^3\). When we set \(y = 0\) and find \(y^{\prime\prime\prime}(0) = 0\), determining stability becomes complex.

In deeper mathematical analysis, a test like linearization could offer further insights, but in this exercise, it's enough to recognize that sometimes, direct stability assessment via derivatives offers limited information. Nonetheless, broader stability analysis could involve numerical simulations or additional mathematical tools.

Phase Plane Trajectories

Phase plane trajectories offer a graphical way to visualize how solutions to differential equations evolve over time. We employ a phase plane with \(y\) against \(y'\) to see the system's dynamics. For our function, converting the equation \(y'' + y^3 = 0\) using \(v = y'\) yields \(v' = -y^3\). This transformation helps align the equations for phase plane plotting.

With this setup, tools such as Python or Desmos can plot these trajectories. The plotted paths depict how \(y\) and \(y'\) change, illustrating behaviors around the equilibrium. These trajectories provide intuitive insights into the possible behaviors of our system, such as oscillations, convergence, or divergence from equilibrium.