// Numbas version: exam_results_page_options {"name": "Harry's copy of 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": [{"statement": "
The thickness of ice on water, $x(t)$, grows according to the equation:
\\[\\frac{dx}{dt}=\\simplify[std]{{a}/(x+{b})^{n}}\\]
Given that $x(0)=0$ find $x(t)$.
On rearranging the equation we get $\\displaystyle{\\simplify[std]{(x+{b})^{n}*(dx/dt) = {a}}}$ and on integrating we obtain:
$\\displaystyle{\\simplify[std]{(x+{b})^{n+1}/{n+1}={a}t +A} \\Rightarrow x+\\var{b}=(A+\\var{a*(n+1)}t)^{1/\\var{n+1}}}$
Using the condition $x(0)=0$ gives $\\displaystyle{A^{1/\\var{n+1}}=\\var{b} \\Rightarrow A=\\var{b^(n+1)}}$
\nHence the solution is:
\\[x(t) = \\simplify[std]{({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}}\\]
The thickness at time $t$ is given by:
\n$x(t)=\\;\\;$[[0]]
\nInput all numbers as integers or fractions – not as decimals.
", "gaps": [{"marks": 3, "vsetrangepoints": 5, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "Input all numbers as integers or fractions.
", "strings": ["."]}, "showpreview": true, "type": "jme", "answer": "({(b ^ (n + 1))} + ({(a * (n + 1))} * t)) ^ (1 / {(n + 1)}) - {b}", "checkingtype": "absdiff", "checkingaccuracy": 0.0001, "scripts": {}, "showCorrectAnswer": true, "answersimplification": "std", "vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false}], "scripts": {}, "showCorrectAnswer": true}], "variables": {"b": {"definition": "random(1..9)", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"definition": "random(1..9)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n": {"definition": "random(1..4)", "name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "metadata": {"notes": "29/06/2012:
\nAdded and edited tags.
\nChecked answer. Checking range OK as we are taking roots of positive numbers, given the choice of ranges for the variables.
\n18/07/2012:
\nAdded description.
\n23/07/2012:
\nAdded tags.
\nThe arbitrary constant A should be relabelled as A_1 in the Advice section part way though the solution.
\nQuestion appears to be working correctly.
\n", "description": "
Solve for $x(t)$, $\\displaystyle\\frac{dx}{dt}=\\frac{a}{(x+b)^n},\\;x(0)=0$
", "licence": "Creative Commons Attribution 4.0 International"}, "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}