Use this tool to sketch the phase plane of a first order system of linear differential equations with constant coefficients[br][br][math]x'=ax+by[/math][br][math]y'=cx+dy[/math][br][br]Enter your coordinates [math]a,b,c[/math] and [math]d[/math] in the left panel, and your phase plane is sketched on the right automatically as eight green phase curves. Eigenvalues and eigenvectors are also automatically calculated, and displayed on the left.[br][br]The moving blue dots indicate the directionality of the solutions as time increases from -1 to 1. To observe the motion over a longer time frame, increase [code]Timeframe[/code] using the slider.

[i]Note 1:[/i] The green phase curves only are plotted for times from -10 to 10 due to a limitation in GeoGebra, and so some systems--such as the system associated with the Owl & Mouse Predator & Prey populations--may not display the phase curves adequately. In particular, the phase curves may be too short to see when the axes have large scales. To get around this, try increasing [code]Timeframe[/code] to observe the dots over a longer period of time even though the phase curves themselves are not visible.[br][br][i]Note 2:[/i] This applet does not currently handle the case of repeated eigenvalues such as described in [url=https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx]this article[/url]. So [math]a=7,b=1,c=-4,d=3[/math] (from Example 1 in the link) will not return a phase portrait. Check back later; I may attempt to code this into an update in the future.

Try replicating Example 1 from [url=https://tutorial.math.lamar.edu/classes/de/phaseplane.aspx]this article on Phase Portraits[/url],[math]a=1,b=2,c=3,d=2[/math].[br][br]For a preview of complex eigenvalues, check out the 2 example systems in [url=https://tutorial.math.lamar.edu/classes/de/ComplexEigenvalues.aspx]this article on complex eigenvalues[/url]. Periodic example: [math]a=3,b=9,c=-4,d=-3[/math]. Note that the eigenvalues are purely complex. Spiral example: [math]a=3,b=-13,c=5,d=1[/math].[br][br]Also, try replicating the Mouse/Owl result from [url=https://www.geogebra.org/m/cxgtwkqa#material/jumcyzry]earlier[/url]. Pay close attention to which model we chose to be [math]x[/math] and which model we chose to be [math]y[/math]! Remember, we have to respect this decision! I'll let you figure out [math]a,b,c[/math] and [math]d[/math]. Take your time. Also, read [i]Note 1[/i] above for a tip on what to expect.

This applet is part of a GeoGebra "Book" on Differential Equations. If you like this, you may enjoy the book. Check it out here:[br][br][url=https://www.geogebra.org/m/cxgtwkqa]https://www.geogebra.org/m/cxgtwkqa[/url]