// Numbas version: exam_results_page_options
{"name": "Solution curves and  Vector fields 1 : dy/dx=1-sin(y)", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"prompt": "<p>Vector field for the differential equation \\[\\frac{dy}{dx}=f(x,y)=1-\\sin(y)\\]</p>\n<p>You change the function $f(x,y)$ by modifying the function $f(x,y)$ in the testfield function defined in Extensions and scripts.</p>\n<p>You have to use Javascript notation for powers and standard functions when you define $f(x,y)$ in testfield.</p>\n<p>If you want to see the solution curve through a particular point then drag the point labeled $(x_0,y_0)$ to it.&nbsp;</p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Using Jsxgraph to draw the vector field for a differential equation of the form $\\frac{dy}{dx}=f(x,y)=1-\\sin(y)$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point.&nbsp;</p>\n<p>If you want to modify $f(x,y)$ simply change the definition of &nbsp;$f(x,y)$ and that of the variable &nbsp;str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.</p>"}, "variable_groups": [], "tags": [], "functions": {"testfield": {"parameters": [], "language": "javascript", "definition": "\n        var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[-5,5,5,-5],grid:true,axis:true});\n      function f(x,y){return 1-Math.sin(y);}\n        var str = \"1-Math.sin(y)\";\n      var board = div.board;\n      for(var i=-10;i<10;i++){\n        for(var j=-10;j<10;j++){\n         //board.create('point',[i/2,j/2],{size:2,name:'',face:'+'});\n      var len=4;\n        // the length of a segment is 1/4\n         var d=1/(len*(Math.pow(1+Math.pow(f(i/2,j/2),2),0.5)));\n       board.create('segment',[[i/2,j/2],[i/2+d,d*f(i/2,j/2)+j/2]]);}\n      }\n        //var N = board.create('slider',[[-4,4],[4,4],[-5,2.5,5]], {name:'N'});\n        var P = board.create('point',[0,0], {name:'(x_0,y_0)'});\n         \n        function doIt() {\n      var txt = JXG.GeonextParser.geonext2JS(str);\n      f = new Function(\"x\", \"yy\", \"var y = yy[0]; var z = \" + txt + \"; return [z]\");\n      board.update();\n        }\n         \n        function ode() {\n       return JXG.Math.Numerics.rungeKutta('heun', [P.Y()], [P.X(), 5], 200, f);\n        }\n        function ode1() {\n       return JXG.Math.Numerics.rungeKutta('heun', [P.Y()], [P.X(), -5], 200, f);\n        } \n        var g = board.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2});\n        g.updateDataArray = function() {\n        var data = ode();\n        var data1= ode1();\n        var h = (5-P.X())/200;\n        var h1 = (5+P.X())/200;\n        this.dataX = [];\n        this.dataY = [];\n        for(var i=0; i<data1.length; i++) {\n        this.dataX[i] = -5+i*h1;\n        this.dataY[i] = data1[data1.length-i-1][0];\n        }\n        for(var i=data1.length; i<data1.length+data.length; i++) {\n        this.dataX[i] = P.X()+(i-data1.length)*h;\n        this.dataY[i] = data[i-data1.length][0];\n        }\n        };\n        doIt();\n      return div;\n        \n        \n      ", "type": "html"}}, "preamble": {"js": "", "css": ""}, "rulesets": {}, "ungrouped_variables": ["a"], "advice": "<p>If you want to see the solution curve through a particular point then drag the point labeled $(x_0,y_0)$ to it.</p>\n<p>The general solution is given by \\[y=x-2\\arctan\\left(1-\\frac{2}{x+C}\\right)\\] for $C$ an arbitrary constant.</p>\n<p>The solution through $(0,0)$ is given by $C=2$.</p>", "variables": {"a": {"group": "Ungrouped variables", "definition": "1", "templateType": "anything", "name": "a", "description": "<p>Filler variable &nbsp;- not used.</p>"}}, "statement": "<p>{testfield()}</p>\n<p></p>", "extensions": ["jsxgraph"], "name": "Solution curves and  Vector fields 1 : dy/dx=1-sin(y)", "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}