Phase portraits for systems of differential equations with complex Eigenvalues

Now we want to look at the phase portraits of systems with complex Eigenvalues. The equilibrium of a system with complex Eigenvalues that have no real part is a stable center around which the trajectories revolve, without ever getting closer to or further from equilibrium. The equilibrium of a system with complex Eigenvalues with a positive real part is an unstable spiral that repels all trajectories. The equilibrium of a system with complex Eigenvalues with a negative real part is an asymptotically stable spiral that attracts all trajectories.

Learn mathKrista KingFebruary 22, 2024math, learn online, online course, online math, differential equations, systems of differential equations, phase portraits, complex eigenvalues, trajectories, center, spiral, stable center, unstable repeller spiral, asymptotically stable attractor spiral

How to graph circles using the center and radius

Given the equation of a circle, we can put the equation in standard form, find the center and radius of the circle from the standard form, and then use the center and radius to graph the circle. Alternately, given the graph of the circle, we can identify the center and radius from the graph and then plug those values into the standard equation of the circle in order to get its equation.

Learn mathKrista KingSeptember 3, 2020math, learn online, online course, online math, algebra, algebra 2, algebra ii, circles, graphing circles, radius of the circle, center, radius, standard equation of the circle, standard form of the equation of a circle, equation of the circle, circle equation, center of the circle

How to find the center, radius, and equation of the sphere

To find the radius, it’s important that we take the square root of the right-hand side, and not just the full value from the right

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