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Question

# Show that the system$\begin{array}{l}{x^{\prime}=y} \\ {y^{\prime}=-\sin x-y}\end{array}$has an equilibrium point at the origin. Compute the Jacobian, then discuss the type and stability of the equilibrium point. Find and describe other equilibria if they exist.

Solution

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In this exercise we will use $\textit{Jacobian}$ of the system, which is defined as:

If we have the linear system

\begin{align*}\begin{cases}x'=f(x,y)\\y'=g(x,y)\end{cases}\end{align*}

its $\textit{Jacobian}$ is the matrix

\begin{align*}\textbf{J}=\begin{bmatrix}f_x(x,y)&f_y(x,y)\\g_x(x,y)&g_y(x,y)\end{bmatrix}\end{align*}

which can help us to discuss the type and stability of the equilibrium point.

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