// Numbas version: exam_results_page_options {"name": "Magda's copy of Resonant 2nd order ODE with exp'l RHS", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Magda's copy of Resonant 2nd order ODE with exp'l RHS", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

For the second-order differential equation \\[\\frac{d^2 z}{dt^2}-\\var{c}\\frac{dz}{dt}+\\var{d}z=\\var{a}e^{\\var{b}t},\\] find the particular integral (response to forcing).

", "advice": "

Firstly we consider the solution to the ODE,

\n

\\[ \\frac{d^2 z}{dt^2}-\\var{c}\\frac{dz}{dt}+\\var{d}z=0\\,.\\]

\n

Using the general form of $z(t) = e^{\\lambda t }$. This gives us the following quadratic,

\n

\\[ \\lambda^2 - \\var{c}\\lambda + \\var{d} = 0 \\]

\n

This has roots $\\lambda = \\var{f}$ and $\\lambda = \\var{b}$. The general solution is therefore,

\n

\\[ z(t) = Ae^\\simplify{{f}t} + B e^{\\var{b} t}.\\] 

\n

Using the RHS of our ODE we see that this corresponds to the second half of our general solution. We can now assume that the particular integral will be of the form,

\n

\\[z_\\mathrm{PI} = C t e^{\\var{b} t}\\,. \\]

\n

Differentiating,

\n

\\[\\frac{dz_\\mathrm{PI}}{dt} = C e^{\\var{b} t} +\\var{b}  C t e^{\\var{b} t}\\,,\\]

\n

whilst 

\n

\\[\\frac{d^2z_\\mathrm{PI}}{dt^2} = \\var{b} C e^{\\var{b} t} + \\var{b}  C e^{\\var{b} t} +\\simplify{{b}^2}  C t e^{\\var{b} t} = \\simplify{2{b}} C e^{\\var{b} t} + \\simplify{{b}^2}  C t e^{\\var{b} t}.\\] 

\n

Subtituting these formulae back into the original ODE gives us,

\n

\\[ \\simplify{2{b}} C e^{\\var{b} t} + \\simplify{{b}^2}  C t e^{\\var{b} t}  - \\var{c}(C e^{\\var{b} t} +\\var{b}  C t e^{\\var{b} t}) +\\var{d}(C t e^{\\var{b} t}) = \\var{a}e^{\\var{b}t}  \\] 

\n

\\[\\Longrightarrow \\simplify{2{b} - {c} }C = \\var{a}  \\] 

\n

\\[\\Longrightarrow C = \\simplify{{a}/(2{b} - {c})}  \\] 

\n

Therefore the particular solution is:

\n

\\[ z_\\mathrm{PI} = \\simplify{{a}/(2{b} - {c}) t} e^{\\var{b} t}.\\]

", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 8#2)", "description": "", "templateType": "randrange"}, "tol1": {"name": "tol1", "group": "Ungrouped variables", "definition": "0.99", "description": "", "templateType": "number"}, "tol2": {"name": "tol2", "group": "Ungrouped variables", "definition": "1.01", "description": "", "templateType": "number"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "b+f", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "b*f", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1 .. 7#2)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "tol1", "tol2", "c", "d", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$z_\\mathrm{PI} $= [[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a}/({b}-{f}))*t*e^({b}*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Paul Bushby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/744/"}, {"name": "George Stagg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/930/"}, {"name": "John Appleby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5728/"}, {"name": "Jordan Day", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9361/"}, {"name": "Magda Carr", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/10043/"}]}]}], "contributors": [{"name": "Paul Bushby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/744/"}, {"name": "George Stagg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/930/"}, {"name": "John Appleby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5728/"}, {"name": "Jordan Day", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9361/"}, {"name": "Magda Carr", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/10043/"}]}