Differentiate
\n$\\simplify{{expression(function)}}$
", "variableReplacementStrategy": "originalfirst"}, {"vsetRangePoints": 5, "showCorrectAnswer": true, "checkVariableNames": false, "marks": 1, "failureRate": 1, "showFeedbackIcon": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "answer": "{expression(correctAns)}", "checkingAccuracy": 0.001, "showPreview": true, "expectedVariableNames": [], "checkingType": "absdiff", "customMarkingAlgorithm": "correctVars:\n set(\"t\")\n\nargument:\n \"t\"\n\nequation:\n d(string(studentExpr),argument,2)+\"-(\"+root1+\"+\"+root2+\")(\"+d(string(studentExpr),argument,1)+\")+\"+(root1*root2)+\"*(\"+string(studentExpr)+\")\"\n\nagree (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n try(\n resultsEqual(unset(question_definitions,eval(parse(equation),vars)),unset(question_definitions,eval(parse(\"0\"),vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\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\nunexpectedVariables:\n if (\"x\" in list(studentVariables),\n incorrect(\"$x$ is the function $x(t)$. We shouldn't have $x$ in our final answer.\"),\n false\n )\n\nsameVars:\n if (not (\"t\" in list(studentVariables)),\n incorrect(\"Our parameter is $t$; your solution does not have $t$ in it.\"),\n true\n )\n\nmark:\n apply(studentExpr);\n apply(unexpectedVariables);\n apply(sameVars);\n apply(numericallyCorrect);\n apply(failMinLength);\n apply(failMaxLength);\n apply(forbiddenStringsPenalty);\n apply(requiredStringsPenalty)", "type": "jme", "vsetRange": [0, 1], "scripts": {}, "variableReplacements": [], "prompt": "Find the general solution, $x(t)$, to
\n$\\simplify{x''-{root1+root2}*x'+{root1*root2}*x}=0$
", "variableReplacementStrategy": "originalfirst"}], "rulesets": {}, "statement": "", "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International", "description": ""}, "tags": [], "ungrouped_variables": [], "variables": {"fns": {"templateType": "anything", "description": "", "name": "fns", "group": "Differential Testing", "definition": "[\"sin\",\"cos\",\"tan\",\"arcsin\",\"arccos\",\"arctan\",\"sinh\",\"cosh\",\"tanh\",\"arcsinh\",\"arccosh\",\"arctanh\",\"ln\",\"sqrt\"]"}, "fnchoice": {"templateType": "anything", "description": "", "name": "fnchoice", "group": "Differential Testing", "definition": "[random(0..length(fns)-1),random(0..length(fns)-1)]"}, "correctAns": {"templateType": "anything", "description": "", "name": "correctAns", "group": "DE Testing", "definition": "if(root1=root2,\"(A+B*t)*e^(\"+root1+\"t)\",\"A*e^(\"+root1+\"t)+B*e^(\"+root2+\"t)\")"}, "root1": {"templateType": "anything", "description": "", "name": "root1", "group": "DE Testing", "definition": "random(-5..5 except 0)"}, "root2": {"templateType": "anything", "description": "", "name": "root2", "group": "DE Testing", "definition": "random(-5..5)"}, "xpow": {"templateType": "anything", "description": "", "name": "xpow", "group": "Differential Testing", "definition": "random(2..5)"}, "function": {"templateType": "anything", "description": "", "name": "function", "group": "Differential Testing", "definition": "fns[fnchoice[0]]+\"(x^\"+xpow+\"*\"+fns[fnchoice[1]]+\"(\"+xcoeff+\"x\"+\"+\"+const+\"))\""}, "xcoeff": {"templateType": "anything", "description": "", "name": "xcoeff", "group": "Differential Testing", "definition": "random(-5..5 except 0)"}, "const": {"templateType": "anything", "description": "", "name": "const", "group": "Differential Testing", "definition": "random(-5..5)"}}, "advice": "", "type": "question"}, {"name": "Differential Equation Question Type", "extensions": ["Differentiation"], "custom_part_types": [{"source": {"pk": 29, "author": {"name": "Andrew Iskauskas", "pk": 724}, "edit_page": "/part_type/29/edit"}, "name": "Differential Equation", "short_name": "differential-equation", "description": "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 characteristic equation is given by trying the solution $x(t)=e^{\\lambda t}$: here this gives
\n$\\simplify{lambda^(2)*e^(lambda*t)+{-root1-root2}*lambda*e^(lambda t)+{root1*root2}*e^(lambda t)}=0.$
\nSince $e^{\\lambda t}>0$ for all $t\\in(-\\infty,\\infty)$, to solve this we must have
\n$\\simplify{lambda^(2)+{-root1-root2}lambda+{root1*root2}}=0$
\nwhich means that we have a repeated root, $\\lambda=\\var{root1}$. To have two linearly independent solutions, we must therefore take $x_{1}(t)=e^{\\var{root1}t}$ and $x_{2}(t)=te^{\\var{root1}t}$. Then the general solution is given by
\n$x(t)=\\simplify{(A+B*t)*e^({root1}*t)},$
\nwhich means that $\\lambda=\\var{root1}$ or $\\lambda=\\var{root2}$. This gives two linearly independent solutions: $x_{1}(t)=e^{\\var{root1}t}$ and $x_{2}(t)=e^{\\var{root2}t}$. So the general solution is
\n$x(t)=\\simplify{A*e^({root1}*t)+B*e^({root2}*t)},$
\nwhere $A$ and $B$ are undetermined constants of integration.
", "variables": {"root1": {"definition": "random(-5..5)", "name": "root1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "root2": {"definition": "random(-5..5)", "name": "root2", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "sol": {"definition": "if(root1=root2,\"(A+B*t)*e^(\"+root1+\"t)\",\"A*e^(\"+root1+\"t)+B*e^(\"+root2+\"t)\")", "name": "sol", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "parts": [{"variableReplacements": [], "type": "differential-equation", "customMarkingAlgorithm": "", "prompt": "Find the general solution to the differential equation for $x(t)$:
\n$\\simplify{x''(t)+{-root1-root2}x'(t)+{root1*root2}*x(t)}=0$
", "showFeedbackIcon": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "settings": {"decoeffs": ["1", "{-(root1+root2)}", "{root1*root2}"], "arg": "t", "sample": "{simplify(expression(sol),'all')}", "var": "x"}, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": "2"}], "statement": "This question concerns finding solutions to a second-order constant-coefficient ODE.
", "preamble": {"css": "", "js": ""}, "functions": {}, "tags": ["characteristic equation", "differential equation", "differentiation"], "rulesets": {}, "variablesTest": {"maxRuns": 100, "condition": "not (root1=0 and root2=0)"}, "metadata": {"description": "Solving a second-order constant coefficient ODE. Uses the differentiation extension: https://github.com/Tandethsquire/Differentiation, and the Differential Equation custom part type, to differentiate a student answer and ensure it satisfies the equation.
", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "type": "question"}]}], "navigation": {"onleave": {"action": "none", "message": ""}, "allowregen": true, "reverse": true, "browse": true, "showresultspage": "oncompletion", "showfrontpage": true, "preventleave": true}, "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International", "description": "A sample of what can be done with the differentiation extension, and the corresponding custom part type.
"}, "name": "Differentiation Extension Examples", "type": "exam", "contributors": [{"name": "Andrew Iskauskas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/724/"}], "extensions": ["Differentiation"], "custom_part_types": [{"source": {"pk": 29, "author": {"name": "Andrew Iskauskas", "pk": 724}, "edit_page": "/part_type/29/edit"}, "name": "Differential Equation", "short_name": "differential-equation", "description": "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.