{"id":423,"date":"2023-02-09T20:58:57","date_gmt":"2023-02-09T20:58:57","guid":{"rendered":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/?page_id=423"},"modified":"2023-11-09T05:57:36","modified_gmt":"2023-11-09T05:57:36","slug":"visualizing-differential-equations","status":"publish","type":"page","link":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/visualizing-differential-equations\/","title":{"rendered":"Visualizing Topics in Differential Equations with Interactive Figures"},"content":{"rendered":"<h1>Visualizing Differential Equations<\/h1>\n<p>Visual exploration can add significant meaning and promote a more general and intuitive understanding of many topics in differential equations.\u00a0 When students can visualize the concepts they are learning, they will understand these concepts more deeply, and find it easier to remember them and connect them to related topics.\u00a0 In differential equations, we have a number of opportunities to help our students visualize the topics we teach.<\/p>\n<p>This page is designed to highlight some of the available resources on this site that can be used in teaching and learning Differential Equations.<\/p>\n<h2>Exploration Apps<\/h2>\n<ul>\n\t<li><strong><a href=\"https:\/\/c3d.libretexts.org\/DirectionField\/index.html\" target=\"_blank\" rel=\"noreferrer noopener\">Direction Field Explorer<\/a><\/strong> <br>\nThis app will plot the direction field for any first-order differential equation that can be solved for $y&#8217;$.\u00a0 A general solution can then be entered with a parameter $C$ and this parameter can then be varied to show how the solution curves will always follow the flow of the direction field vectors (or slope segments of the slope field, if this option is specified). The variables can be adjusted to match the application being studied for many different common variable names.\u00a0 Clicking on the direction field will display a numerically generated solution curve through the selected point.<\/li>\n\t<li><strong><a href=\"https:\/\/c3d.libretexts.org\/CalcPlot3D\/index.html?type=vectorfield;vectorfield=vf;m=x-xy;n=x-y;p=0;visible=true;view=undefined;scale=4;nx=9;ny=9;nz=1;mode=0;twod=true;constcol=true;color=rgb(0,0,255);norm=false;desystem=true&amp;type=slider;slider=t;value=0;steps=100;pmin=0;pmax=5;repeat=true;bounce=false;waittime=1;careful=false;noanimate=false;name=-1&amp;type=window;hsrmode=3;nomidpts=true;anaglyph=-1;center=0,0,10,1;focus=0,0,0,1;up=0,2,0,1;transparent=false;alpha=140;twoviews=false;unlinkviews=false;axisextension=0.7;shownormals=false;shownormalsatpts=false;xaxislabel=x;yaxislabel=y;zaxislabel=z;edgeson=true;faceson=true;showbox=true;showaxes=true;showticks=true;perspective=false;centerxpercent=0.5;centerypercent=0.5;rotationsteps=30;autospin=true;xygrid=false;yzgrid=false;xzgrid=false;gridsonbox=true;gridplanes=false;gridcolor=rgb(128,128,128);xmin=-2;xmax=2;ymin=-2;ymax=2;zmin=-2;zmax=2;xscale=1;yscale=1;zscale=1;zcmin=-4;zcmax=4;xscalefactor=1;yscalefactor=1;zscalefactor=1;tracemode=0;keep2d=false;zoom=1.236\" target=\"_blank\" rel=\"noreferrer noopener\">CalcPlot3D<\/a><\/strong> <br>\nThis app allows the exploration of the phase portraits and solutions of systems of differential equations, both linear and non-llinear, and both autonomous and non-autonomous.\u00a0 That is, you can even include a time parameter $t$ in the equations in your system.\u00a0 Using parameter animation, CalcPlot3D will even show how the corresponding direction field changes as time passes and will display a corresponding solution curve when the user clicks on the plot.<br>\n<br>\nSince we spend more time on autonomous systems, let&#8217;s discuss what else can be done with them.\u00a0 In addition to plotting the phase portrait, the user can Add a Space Curve to the phase portrait plot to represent the parametric form of the general solution of the system of differential equations.\u00a0 In CalcPlot3D we can select from the parameters $a, b, c,$ and $d$ in CalcPlot3D to represent the parameters in the general solution.\u00a0 Once this space curve is plotted on the phase portrait, the included parameters can be varied with sliders (or textboxes) to confirm that the solution curves always follow the vectors in the phase portrait.<\/li>\n<\/ul>\n<h2>Supplementary Textbook Content<\/h2>\n<ul>\n\t<li><a href=\"https:\/\/math.libretexts.org\/Courses\/Monroe_Community_College\/MTH_225_Differential_Equations\/2%3A_First_Order_Equations\/2.4%3A_Bifurcations\" target=\"_blank\" rel=\"noreferrer noopener\">A Visual Introduction to Bifurcation, <strong>Section 2.4 Bifurcation<\/strong><\/a> in Prof. Seeburger&#8217;s custom OER Differential Equations textbook on the LibreTexts platform<\/li>\n\t<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=yDdW-1mlETg&amp;list=PL83Q_gTbFatSHn7YXfD0bY9smoxolzqWf\" target=\"_blank\" rel=\"noreferrer noopener\">Video presentation<\/a><\/strong> by Prof. Seeburger about customizing an OER textbook for Differential Equations on the LibreTexts platform.<\/li>\n<\/ul>\n<h2>GeoGebra Simulations and Standalone Interactive Figures<\/h2>\n<ul>\n\t<li>\n<h3>Spring Motion Simulation<\/h3>\n<ul>\n\t<li><strong><a href=\"https:\/\/www.geogebra.org\/m\/ykqc5be6\" target=\"_blank\" rel=\"noreferrer noopener\">Spring Motion Simulation (with\/without Forcing)<\/a><\/strong><\/li>\n<\/ul>\n<\/li>\n\t<li>\n<h3>Bifurcation Motivation<\/h3>\n<ul>\n\t<li><a href=\"https:\/\/www.geogebra.org\/m\/ubrrxnx8\" target=\"_blank\" rel=\"noreferrer noopener\">Example 1<\/a>, \\(\\dfrac{dy}{dt} = y^2 + 2y + a\\),\u00a0 Dynamic graph of $f(y) = y&#8217;$ to illustrate Bifurcation<\/li>\n<\/ul>\n<\/li>\n\t<li>\n<h3>Bifurcation Diagrams<\/h3>\n<ul>\n\t<li><a href=\"https:\/\/www.geogebra.org\/m\/eqtydqdq\" target=\"_blank\" rel=\"noreferrer noopener\">Example 1<\/a>, $\\dfrac{dy}{dt} = y^2 + 2y + a$,\u00a0 Bifurcation Diagram<\/li>\n\t<li><a title=\"Opens external link in new window\" href=\"https:\/\/www.geogebra.org\/m\/jufntewh\" target=\"_blank\" rel=\"noreferrer noopener\">Bifurcation diagram for: $\\dfrac{dy}{dt} =\\frac{1}{50}(y^2 + ay + 16)$<\/a><\/li>\n<\/ul>\n<\/li>\n\t<li>\n<h3>Laplace Transforms<\/h3>\n<ul>\n\t<li><a href=\"https:\/\/www.geogebra.org\/m\/usn9espf\" target=\"_blank\" rel=\"noreferrer noopener\">Interactive Piecewise Function Plotter with the Unit Step Function<\/a><\/li>\n<\/ul>\n<\/li>\n\t<li>\n<h3>Phase Portraits for Systems of Differential Equations<\/h3>\n<ul>\n\t<li><a href=\"https:\/\/www.geogebra.org\/m\/xzshp8u5\" target=\"_blank\" rel=\"noreferrer noopener\">Example 1, Saddle Point<\/a><\/li>\n\t<li><a href=\"https:\/\/www.geogebra.org\/m\/nes6uxwf\" target=\"_blank\" rel=\"noreferrer noopener\">DE System Explorer<\/a>, allows you to see the phase portrait of any 2 x 2 linear system of ODEs.\u00a0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>See the <strong><a href=\"https:\/\/sites.monroecc.edu\/multivariablecalculus\/geogebra-visualizations\/\" target=\"_blank\" rel=\"noreferrer noopener\">GeoGebra Apps<\/a><\/strong> page for more Interactive Figures designed to help visualize Calculus and Differential Equations.<\/p>","protected":false},"excerpt":{"rendered":"<p>Visualizing Differential Equations Visual exploration can add significant meaning and promote a more general and intuitive understanding of many topics in differential equations.\u00a0 When students can visualize the concepts they are learning, they will understand these concepts more deeply, and find it easier to remember them and connect them to related topics.\u00a0 In differential equations, [&#8230;]<\/p>\n","protected":false},"author":654,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-423","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/pages\/423","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/users\/654"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/comments?post=423"}],"version-history":[{"count":8,"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/pages\/423\/revisions"}],"predecessor-version":[{"id":472,"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/pages\/423\/revisions\/472"}],"wp:attachment":[{"href":"https:\/\/sites.monroecc.edu\/multivariablecalculus\/wp-json\/wp\/v2\/media?parent=423"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}