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{"name": "Transform second order ODE into 2D and 3D dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}}, "ungrouped_variables": ["a1", "c1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Transform second order ODE into 2D and 3D dynamical systems", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-b1}*x-{a1}*y-{c1}*sin(t)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>By setting $y=\\dot{x}$, transform the differential equation into a 2-dimensional non-autonomous dynamical system of the form</p>\n        <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,t),\\\\\\dot{y}&amp;=g(x,y,t).\\end{align}\\]</p>\n        <p>$f(x,y,t)=$ [[0]].</p>\n        <p>$g(x,y,t)=$ [[1]].</p>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-b1}*x-{a1}*y-{c1}*sin(z)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>By setting $y=\\dot{x}$ and $z=t$, transform the differential equation into a 3-dimensional autonomous dynamical system of the form</p>\n        <p>\\[\\begin{align}\\dot{x}&amp;=f(x,y,z),\\\\\\dot{y}&amp;=g(x,y,z),\\\\\\dot{z}&amp;=h(x,y,z).\\end{align}\\]</p>\n        <p>$f(x,y,z)=$ [[0]].</p>\n        <p>$g(x,y,z)=$ [[1]].</p>\n        <p>$h(x,y,z)=$ [[2]].</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Consider the following second order differential equation:</p>\n  <p>\\[\\simplify[std]{ddot:x+{a1}*dot:x+{b1}*x+{c1}*sin(t)}=0,\\]</p>\n  <p>where $\\dot{x}$ denotes the derivative of $x$ with respect to $t$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Transform a second order ODE into 2D non-autonomous and 3D autonomous dynamical systems of ODEs.</p>"}, "advice": "<p>a)</p>\n  <p>Let $y=\\dot{x}$, then $\\ddot{x}=\\dot{y}$, and the equation becomes</p>\n  <p>\\[\\simplify[std]{dot:y+{a1}*y+{b1}*x+{c1}*sin(t)}=0,\\]</p>\n  <p>and so</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,t),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*x-{a1}*y-{c1}*sin(t)}=g(x,y,t).\\end{align}\\]</p>\n  <p></p>\n  <p>b)</p>\n  <p>Now also let $z=t$, and so $\\dot{z}=1$, then</p>\n  <p>\\[\\begin{align}\\dot{x}&amp;=y=f(x,y,z),\\\\\\dot{y}&amp;=\\simplify[std]{{-b1}*x-{a1}*y-{c1}*sin(z)}=g(x,y,z),\\\\\\dot{z}&amp;=1=h(x,y,z).\\end{align}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}