// Numbas version: exam_results_page_options
{"name": "2 ODE Eq 1 [Explore]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2 ODE Eq 1 [Explore]", "tags": [], "metadata": {"description": "<p>Practice question solving linear homogeneous second order differentials using the auxiliary equation method.</p>", "licence": "All rights reserved"}, "statement": "<p>Solve the following second order differential</p>\n<p></p>\n<p>$\\frac{d^{2}y}{dx^{2}}\\var{a}\\frac{dy}{dx}\\var{b}y=0$ where $y=\\var{c}$ and $\\frac{dy}{dx}=2$ when $x=0$.</p>", "advice": "<p>$\\frac{d^{2}y}{dx^{2}}\\var{a}\\frac{dy}{dx}\\var{b}y=0$ where $y=\\var{c}$ and $\\frac{dy}{dx}=2$ when $x=0$.</p>\n<p>Simplify the differential by substituting $D=\\frac{d}{dx}$ and extract $y$</p>\n<p>$(D^{2}\\var{a}D\\var{b})y=0$&nbsp;</p>\n<p>Convert the terms between the brackets to an auxiliary equation by replacing $D$ with $m$ to give</p>\n<p>$(m^{2}\\var{a}m\\var{b})y=0$&nbsp;</p>\n<p>Find the roots of m usign the quadratic roots formula</p>\n<p>\\[\\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}&nbsp; \\]</p>\n<p>Where $a=1$, $b=\\var{a}$ and $c=\\var{b}$</p>\n<p>\\[\\frac{-\\var{a} \\pm \\sqrt{\\var{a}^2-4(1)(\\var{b})}}{2}&nbsp; \\]</p>\n<p>Roots: m = \\[ \\var{Rootalpha} \\] or \\[ \\var{Rootbeta} \\]</p>\n<p>Roots are Real and Different</p>\n<p>General solution will be in the form $y = Ae^{\\alpha x}+Be^{\\beta x}$</p>\n<p>Let the roots of m equal $\\alpha$ and $\\beta$</p>\n<p>$\\alpha = \\var{Rootalpha}$ and $\\beta = \\var{RootBeta}$</p>\n<p></p>\n<p>$y = Ae^{\\var{Rootalpha}x}+Be^{\\var{Rootbeta}x}$...general solution</p>\n<p>To find the particular solution, sub in the inital conditions: $y=\\var{c}$ and $\\frac{dy}{dx}=2$ when $x=0$.</p>\n<p>$\\var{c} = Ae^{\\var{Rootalpha}(0)}+Be^{\\var{Rootbeta}(0)}$</p>\n<p>$\\var{c} = A+ B$ ... eq(1)</p>\n<p>Stille have 2 unkwons, A and B. To use the other initial condition, $\\frac{dy}{dx}=2$ when $x=0$, the general solution must be differentiated.</p>\n<p>\\[\\frac{dy}{dx} = \\var{Rootalpha}Ae^{\\var{Rootalpha}x}+\\var{Rootbeta}Be^{\\var{Rootbeta}x} \\]</p>\n<p>Sub in the conditions</p>\n<p>\\[2 = \\var{Rootalpha}Ae^{\\var{Rootalpha}(0)}+\\var{Rootbeta}Be^{\\var{Rootbeta}(0)} \\]</p>\n<p>\\[2 = \\var{Rootalpha}A+\\var{Rootbeta}B \\]&nbsp; ....eq(2)</p>\n<p>Let $B=\\var{c}-A$ .... from eq(1)</p>\n<p>Sub into eq(2)</p>\n<p>\\[2 = \\var{Rootalpha}A+\\var{Rootbeta}(\\var{c}-A) \\]</p>\n<p>\\[ A = \\frac{2-(\\var{Rootbeta})(\\var{c})}{\\var{Rootalpha}-\\var{Rootbeta}}= \\var{(2-({Rootbeta})({c}))/({Rootalpha}-{Rootbeta})} \\]</p>\n<p></p>\n<p>Sub A into eq(1)</p>\n<p>\\[ \\var{c} = \\var{(2-({Rootbeta})({c}))/({Rootalpha}-{Rootbeta})} + B \\]</p>\n<p>\\[ B= \\var{{c}- (2-({Rootbeta})({c}))/({Rootalpha}-{Rootbeta})} \\]</p>\n<p>Sun back into the general solution</p>\n<p>\\[y = \\var{(2-({Rootbeta})({c}))/({Rootalpha}-{Rootbeta})}e^{\\var{Rootalpha}x}+\\var{{c}- (2-({Rootbeta})({c}))/({Rootalpha}-{Rootbeta})}e^{\\var{Rootbeta}x} \\]...particular solution</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9 .. -1#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5 .. -1#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "Rootalpha": {"name": "Rootalpha", "group": "Ungrouped variables", "definition": "precround({(-a+((a^2-4*b)^(1/2)))/2},2)", "description": "", "templateType": "anything", "can_override": false}, "Rootbeta": {"name": "Rootbeta", "group": "Ungrouped variables", "definition": "precround({(-a-((a^2-4*b)^(1/2)))/2},2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "Rootalpha", "Rootbeta"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Auxiliary Equation", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "General Solution Form", "rawLabel": "", "otherPart": 1, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Evaluate the roots for the equation given.</p>\n<p>$\\alpha =$ [[0]]&nbsp; &nbsp;and&nbsp; &nbsp; $\\beta =$ [[1]]</p>\n<p></p>\n<p>Hint: assign alpha to the answer of $-b+\\sqrt{}$ and $\\beta$ to the $-b - \\sqrt{}$</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 0.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.95*{(-{a}+({a}^2-4{b})^(1/2))/2}", "maxValue": "1.05*{(-{a}+({a}^2-4{b})^(1/2))/2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.95*{(-{a}-({a}^2-4{b})^(1/2))/2}", "maxValue": "1.05*{(-{a}-({a}^2-4{b})^(1/2))/2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": true, "customName": "General Solution Form", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "General Solution", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What type of roots are these?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Real and Different", "Real and Equal", "Complex"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}, {"type": "gapfill", "useCustomName": true, "customName": "General Solution", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Particular Solution", "rawLabel": "", "otherPart": 3, "variableReplacements": [], "availabilityCondition": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>State the general solution to the equation using the roots&nbsp;</p>\n<p>\\[\\alpha = \\var{Rootalpha}\\] and \\[ \\beta = \\var{Rootbeta}\\]</p>\n<p>y=[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 2.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "A*e^({Rootalpha}*x)+B*e^({Rootbeta}*x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Particular Solution", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Evaluate the constants A and B for the following initial conditions</p>\n<p>$y=\\var{c}$ and $\\frac{dy}{dx}=2$ when $x=0$.</p>\n<p></p>\n<p>$ y =$ [[0]] $e^{\\var{Rootalpha}x}+$ [[1]]$e^{\\var{Rootbeta}x}&nbsp; $</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 3.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.95*((2-{Rootbeta}*{c})/({Rootalpha}-{Rootbeta}))", "maxValue": "1.05*((2-{Rootbeta}*{c})/({Rootalpha}-{Rootbeta}))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Gap 4.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.95*((2-{Rootalpha}*{c})/({Rootbeta}-{Rootalpha}))", "maxValue": "1.05*((2-{Rootalpha}*{c})/({Rootbeta}-{Rootalpha}))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "explore", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}]}], "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}