// Numbas version: exam_results_page_options
{"name": "Separable first order ODE with boundary condition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "b"}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "description": "", "name": "n"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "b", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Separable first order ODE with boundary condition", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>The thickness at time $t$ is given by:</p>\n        <p>$x(t)=\\;\\;$[[0]]</p>\n        <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The thickness of ice on water, $x(t)$, grows according to the equation:<br/>\\[\\frac{dx}{dt}=\\simplify[std]{{a}/(x+{b})^{n}}\\]<br/>Given that $x(0)=0$ find $x(t)$.</p>", "tags": ["1st order differential equation", "Calculus", "checked2015", "differential equation", "differential equation ", "first order differential equation", "growth", "initial conditions", "MAS1603", "modelling", "ode", "separable variables", "separation of variables"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p>Added and edited tags.</p>\n    <p>Checked answer. Checking range OK as we are taking roots of positive numbers, given the choice of ranges for the variables.</p>\n    <p><strong>18/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>The arbitrary constant A should be relabelled as A_1 in the Advice section part way though the solution.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>&nbsp;</strong></p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve for $x(t)$, $\\displaystyle\\frac{dx}{dt}=\\frac{a}{(x+b)^n},\\;x(0)=0$</p>"}, "advice": "<p>On rearranging the equation we get $\\displaystyle{\\simplify[std]{(x+{b})^{n}*(dx/dt) = {a}}}$ and on integrating we obtain:<br/>$\\displaystyle{\\simplify[std]{(x+{b})^{n+1}/{n+1}={a}t +A} \\Rightarrow x+\\var{b}=(A+\\var{a*(n+1)}t)^{1/\\var{n+1}}}$</p>\n    <p>Using the condition $x(0)=0$ gives $\\displaystyle{A^{1/\\var{n+1}}=\\var{b} \\Rightarrow A=\\var{b^(n+1)}}$</p>\n    <p>Hence the solution is:<br/>\\[x(t) = \\simplify[std]{({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}}\\]</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/"}]}