For questions on differentiation. Takes the student answer and explicitly checks that it satisfies a given differential equation.
\nWill also check that the general solution has the correct number of linearly independent solutions (for up to a third-order ODE).
", "help_url": "", "input_widget": "jme", "input_options": {"correctAnswer": "settings['sample']", "hint": {"static": true, "value": ""}, "showPreview": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\napply(unexpectedVariables);\napply(sameVars);\napply(hasConstants);\nif(length(constants)=2 or length(constants)=3, apply(notInd), true);\napply(numericallyCorrect)\n\ninterpreted_answer:\nstring(studentAnswer)\n\nequation:\njoin(map(settings['decoeffs'][n]+\"*(\"+d(string(studentAnswer),settings['arg'],length(settings['decoeffs'])-1-n)+\")\",n,0..length(settings['decoeffs'])-1),'+')\n\nagree:\nmap(\n try(\n resultsEqual(eval(parse(equation),vars),eval(settings['RHS'],vars),'RelDiff',0.0001),\n message,\n warn(translate(\"part.jme.answer invalid\",[\"message\":message]));\n fail(translate(\"part.jme.answer invalid\",[\"message\":message]));\n false\n ),\n vars,\n vset\n )\n\nvset:\nrepeat(\n dict(map([x,random(vRange)],x,correctVars or studentVariables)),\n 5\n)\n\ncorrectVars:\nset(settings['arg'])\n\nstudentVariables:\nset(findvars(studentAnswer))\n\nvRange:\n0..1#0\n\nunexpectedVariables:\nif (settings['var'] in list(studentVariables),\n incorrect(\"$\"+settings['var']+\"$ is the function $\"+settings['var']+\"(\"+settings['arg']+\")$; it should not be in the final answer.\"); end(),\n false\n ) \n\nsameVars:\nif (not (settings['arg'] in list(studentVariables)),\n incorrect(\"The argument of the function is $\"+settings['arg']+\"$; your solution does not contain it.\"); end(),\n true\n )\n\nnumFails:\napply(agree);\nlen(filter(not x,x,agree))\n\nnumericallyCorrect:\napply(numFails);\nif(numFails<1,\n correct(translate('part.jme.marking.correct')),\n incorrect()\n ) \n\nhasConstants:\nif(not(length(studentVariables)=length(findvars(settings[\"sample\"]))),\n incorrect(\"Your solution does not have the required number of constants to be a general solution.\"); end(),\n true\n )\n\nconstants:\nfilter(not(x=settings['arg']),x,map(x,x,studentVariables))\n\ntestSet:\nrepeat([settings['arg'],random(vRange)],5)\n\ntestingSets:\nmap(\nmap(\n dict(\n map([constants[m],if(m=n,1,0)],m,0..length(constants)-1)\n +[vars]),\n n,0..length(constants)-1\n),\nvars,testSet)\n\nlinears:\nmap(\nmap(\n map(eval(parse(d(string(studentAnswer),settings['arg'],m)),vars),m,0..length(constants)-1),\n vars,testingSets[x0]),\nx0,0..length(testingSets)-1)\n\nnotInd:\nif(length(constants)=2 or length(constants)=3,\nif(some(map(det(matrix(linears[n]))=0,n,0..length(linears)-1)),\n incorrect(\"Your solutions are not linearly independent.\"); end(),\n false\n),\nfalse\n)", "marking_notes": [{"description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "apply(unexpectedVariables);\napply(sameVars);\napply(hasConstants);\nif(length(constants)=2 or length(constants)=3, apply(notInd), true);\napply(numericallyCorrect)", "name": "mark"}, {"description": "A value representing the student's answer to this part.", "definition": "string(studentAnswer)", "name": "interpreted_answer"}, {"description": "The differential equation in the question
", "definition": "join(map(settings['decoeffs'][n]+\"*(\"+d(string(studentAnswer),settings['arg'],length(settings['decoeffs'])-1-n)+\")\",n,0..length(settings['decoeffs'])-1),'+')", "name": "equation"}, {"description": "Do the student's answer and the expected answer agree on each of the sets of variable values?
", "definition": "map(\n try(\n resultsEqual(eval(parse(equation),vars),eval(settings['RHS'],vars),'RelDiff',0.0001),\n message,\n warn(translate(\"part.jme.answer invalid\",[\"message\":message]));\n fail(translate(\"part.jme.answer invalid\",[\"message\":message]));\n false\n ),\n vars,\n vset\n )", "name": "agree"}, {"description": "The set of variable values to test against
", "definition": "repeat(\n dict(map([x,random(vRange)],x,correctVars or studentVariables)),\n 5\n)", "name": "vset"}, {"description": "Variables used in the correct answer
", "definition": "set(settings['arg'])", "name": "correctVars"}, {"description": "Variables used in the student's answer
", "definition": "set(findvars(studentAnswer))", "name": "studentVariables"}, {"description": "The range to pick variable values from
", "definition": "0..1#0", "name": "vRange"}, {"description": "Unexpected variables used in the student's answer
", "definition": "if (settings['var'] in list(studentVariables),\n incorrect(\"$\"+settings['var']+\"$ is the function $\"+settings['var']+\"(\"+settings['arg']+\")$; it should not be in the final answer.\"); end(),\n false\n ) ", "name": "unexpectedVariables"}, {"description": "Does the student use the same variables as the correct answer?
", "definition": "if (not (settings['arg'] in list(studentVariables)),\n incorrect(\"The argument of the function is $\"+settings['arg']+\"$; your solution does not contain it.\"); end(),\n true\n )", "name": "sameVars"}, {"description": "The number of times the student's answer and the expected answer disagree
", "definition": "apply(agree);\nlen(filter(not x,x,agree))", "name": "numFails"}, {"description": "Is the student's answer numerically correct?
", "definition": "apply(numFails);\nif(numFails<1,\n correct(translate('part.jme.marking.correct')),\n incorrect()\n ) ", "name": "numericallyCorrect"}, {"description": "Checks that the student's solution has the correct number of undetermined constants, by comparing to the sample solution.
", "definition": "if(not(length(studentVariables)=length(findvars(settings[\"sample\"]))),\n incorrect(\"Your solution does not have the required number of constants to be a general solution.\"); end(),\n true\n )", "name": "hasConstants"}, {"description": "Finds the constants in the student's expression.
", "definition": "filter(not(x=settings['arg']),x,map(x,x,studentVariables))", "name": "constants"}, {"description": "Generates a random set of x values to test independence of solutions over.
", "definition": "repeat([settings['arg'],random(vRange)],5)", "name": "testSet"}, {"description": "Creates a set of 'linearly independent' solutions, by sequentially setting one constant to one and the others to zero.
", "definition": "map(\nmap(\n dict(\n map([constants[m],if(m=n,1,0)],m,0..length(constants)-1)\n +[vars]),\n n,0..length(constants)-1\n),\nvars,testSet)", "name": "testingSets"}, {"description": "Evaluates the 'linearly independent' solutions over a pre-determined sample of argument values.
", "definition": "map(\nmap(\n map(eval(parse(d(string(studentAnswer),settings['arg'],m)),vars),m,0..length(constants)-1),\n vars,testingSets[x0]),\nx0,0..length(testingSets)-1)", "name": "linears"}, {"description": "Checks that the student does have the right number of linearly independent solutions. Fails if any of the determinants are zero: the determinants are the Wronskian W for the 'independent' solutions y1(x)=y(x;1,0,...,0), y2(x)=(x;0,1,...,0),...,yn(x)=(x;0,0,...,1).
\n", "definition": "if(length(constants)=2 or length(constants)=3,\nif(some(map(det(matrix(linears[n]))=0,n,0..length(linears)-1)),\n incorrect(\"Your solutions are not linearly independent.\"); end(),\n false\n),\nfalse\n)", "name": "notInd"}], "settings": [{"hint": "A possible solution, to be shown to the student on 'Reveal Answer'.", "label": "Sample Solution", "help_url": "", "subvars": true, "default_value": "", "name": "sample", "input_type": "mathematical_expression"}, {"hint": "A list of strings corresponding to the (possible function-valued) coefficients of the various terms in the differential equation.The solution to this equation is a function that is itself after twice differentiating. Once such function is $Ae^x$ for any $A\\in\\mathbb{R}$, since this function differentiates to itself immediately. Two further options are $\\cosh(x)$ and $\\sinh(x)$.
\nHence any of these functions (or any linear combination of them) is a solution to the differential equation $y=\\frac{d^2y}{dx^2}$. That is, the general solution is of the form
\n$$
y=Ae^x+B\\cosh(x)+C\\sinh(x), \\quad x \\in \\mathbb{R}.
$$
Suggest a solution to the following differential equation:
\n$$
y=\\frac{d^2y}{dx^2}
$$