The auxiliary equation $P(m) = 0$ of a $5^{\\text{th}}$ order ordinary differential equation of the form $P(D) y = 0$ yields the following solutions:
\n$m_1 = \\var{{a}}$ ;
$m_2 = \\var{{b}}$ ($m_2$ has algebraic multiplicity equal to $2$);
$m_3 = \\var{{c}} + \\sqrt{\\var{{d}}} i$; and
$m_4 = \\var{{c}} - \\sqrt{\\var{{d}}} i$.
What is the differential equation?
$\\Big($ [[0]] $\\Big) \\frac{\\mathrm{d}^5 y}{\\mathrm{d} x^5} + \\Big($ [[1]] $\\Big) \\frac{\\mathrm{d}^4 y}{\\mathrm{d} x^4} + \\Big($ [[2]] $\\Big) \\frac{\\mathrm{d}^3 y}{\\mathrm{d} x^3} + \\Big($ [[3]] $\\Big) \\frac{\\mathrm{d}^2 y}{\\mathrm{d} x^2} + \\Big($ [[4]] $\\Big) \\frac{\\mathrm{d} y}{\\mathrm{d} x} + \\Big($ [[5]] $\\Big) y = 0$.
What is the egneral solution of the differential equation? If you wish to write something like $e^{2x}$ then please write e^(2*x); and if you wish to write something like $\\sqrt{2} x$ then please write ((2)^(1/2))*x .
$y(x) = A$ [[6]] $+(B + Cx)$ [[7]] $+$ [[8]] $\\Big( D \\cos \\Big($ [[9]] $\\Big) + E \\sin \\Big($ [[9]] $\\Big) \\Bigg)$.
Consider the differential equation
$\\simplify{diff(y,x,3) + {B_coeff_2}*diff(y,x,2) + {B_coeff_1}*diff(y,x,1) + {B_coeff_0}*y} = x + e^x$
We will find its general solution.
First, the auxiliary equation of the homogeneous equation is
$\\Big($ [[0]] $\\Big) m^3 + \\Big($ [[1]] $\\Big) m^2 + \\Big($ [[2]] $\\Big) m + \\Big($ [[3]] $\\Big) = 0$
Note that any real solution to this equation will divide the constant term, [[3]].
This has the real solution $r_1 = $ [[4]] and the complex solutions are (please write the complex root with positive imaginary part, first) $r_2 =$ [[5]] and $r_3 =$ [[6]].
\nHence, the complimentary function is
$(CF)(x)$
$= A$ [[7]] $+$ [[8]] $\\Bigg( B \\cos \\Big($ [[9]] $\\Big) + C \\sin \\Big($ [[9]] $\\Big) \\Bigg)$
where $A$,$B$, and $C$ are arbitrary real constants.
Now we will find a particular integral. The right-hand-side of the non-homogeneous equation is $x + e^x$, and so we should try $y(x) = ax + b +ce^x$, where $a,b,$ and $c$ are constants to be found.
\nBy substituting $y(x) = ax + b +ce^x$ into the non-homogeneous equation, and comparing 1) the coefficients of $x$, 2) the coefficients of $e^x$, and 3) the constants, we can deduce the following (please ensure that your answers are in single fractions; e.g. if you wish to write $\\frac{1}{2} + \\frac{1}{3}$ then please write $\\frac{5}{6}$; otherwise, the system will not correctly interpret what you wish to write).
$a =$ [[10]] ;
$b =$ [[11]] ;
$c =$ [[12]] .
So, a particular integral is
$(PI)(x) = \\Big($ [[10]] $\\Big)x + \\Big($ [[11]] $\\Big) + \\Big($ [[12]] $\\Big) e^x$.
Hence, the general solution to the non-homogeneous equation is
$y(x) = (CF)(x) + (PI)(x)$
$= A$ [[7]] $+$ [[8]] $\\Bigg( B \\cos \\Big($ [[9]] $\\Big) + C \\sin \\Big($ [[9]] $\\Big) \\Bigg) + \\Big($ [[10]] $\\Big)x + \\Big($ [[11]] $\\Big) + \\Big($ [[12]] $\\Big) e^x$.
In the remainder of the question, we will look at exact ODEs.
\nConsider a first order differential equation of the form
\n$M(x,y) \\mathrm{d} x + N(x,y) \\mathrm{d} y = 0$
\nwhere $M(x,y)$ and $N(x,y)$ are given functions of $x$ and $y$.
\nThis is an exact ODE if and only if there exists a function $F(x,y)$ such that
[[0]]
This is equivalent to the condition that
[[1]]
Now consider the following first order differential equation:
\n$\\Big( \\simplify{{aa}*{bb}} x^{\\simplify{{bb}-1}} y^{\\var{{cc}}} + \\var{{dd}} \\cos (y) \\Big) \\mathrm{d} x + \\Big( \\simplify{{aa}*{cc}} x^{\\var{{bb}}} y^{\\simplify{{cc}-1}} - (\\var{{dd}} x + \\var{{ff}}) \\sin (y) + \\cos(y) \\Big) \\mathrm{d} y$.
\nThat is, $M(x,y) = \\simplify{{aa}*{bb}} x^{\\simplify{{bb}-1}} y^{\\var{{cc}}} + \\var{{dd}} \\cos (y)$ and $N(x,y) = \\simplify{{aa}*{cc}} x^{\\var{{bb}}} y^{\\simplify{{cc}-1}} - (\\var{{dd}} x + \\var{{ff}}) \\sin (y) + \\cos(y)$.
\nWhat are $\\frac{\\partial M}{\\partial y}$ and $\\frac{\\partial N}{\\partial x}$? (If you wish to write something like $2x^3 y^4$ then please write 2*(x^3)*(y^4); and if you wish to write something like $(2x+3) \\cos(y)$ then please write (2*x + 3)*cos(y) .)
$\\frac{\\partial M}{\\partial y} =$ [[2]]
$\\frac{\\partial N}{\\partial x} =$ [[3]]
Hence, we can see that this ODE is [[4]]
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""], "choices": ["$\\frac{\\partial F}{\\partial x} = M(x,y)$ and $\\frac{\\partial F}{\\partial y} = N(x,y)$.
", "$\\frac{\\partial F}{\\partial y} = M(x,y)$ and $\\frac{\\partial F}{\\partial x} = N(x,y)$.
", "$\\frac{\\partial F}{\\partial y} = M(x,y)$ and $\\frac{\\partial F}{\\partial y} = N(x,y)$.
", "$\\frac{\\partial F}{\\partial x} = M(x,y)$ and $\\frac{\\partial F}{\\partial x} = N(x,y)$.
"], "displayColumns": "1", "showFeedbackIcon": true, "showCorrectAnswer": false, "minMarks": 0, "scripts": {}, "displayType": "radiogroup", "shuffleChoices": true, "unitTests": [], "maxMarks": 0, "customMarkingAlgorithm": "", "type": "1_n_2"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""], "choices": ["$\\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial x}$.
", "$\\frac{\\partial M}{\\partial x} = \\frac{\\partial N}{\\partial x}$.
", "$\\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial y}$.
", "$\\frac{\\partial M}{\\partial x} = \\frac{\\partial N}{\\partial y}$.
"], "displayColumns": "1", "showFeedbackIcon": true, "showCorrectAnswer": false, "minMarks": 0, "scripts": {}, "displayType": "radiogroup", "shuffleChoices": true, "unitTests": [], "maxMarks": 0, "customMarkingAlgorithm": "", "type": "1_n_2"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkVariableNames": false, "answer": "{aa}*{bb}*{cc}*(x^({bb}-1))*(y^({cc}-1)) - {dd}*sin(y)", "unitTests": [], "expectedVariableNames": [], "checkingType": "absdiff", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "type": "jme", "customMarkingAlgorithm": "", "marks": 1}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkVariableNames": false, "answer": "{aa}*{bb}*{cc}*(x^({bb}-1))*(y^({cc}-1)) - {dd}*sin(y)", "unitTests": [], "expectedVariableNames": [], "checkingType": "absdiff", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "type": "jme", "customMarkingAlgorithm": "", "marks": 1}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "matrix": ["1", 0], "distractors": ["", ""], "choices": ["exact.
", "not exact.
"], "displayColumns": 0, "showFeedbackIcon": true, "showCorrectAnswer": false, "minMarks": 0, "scripts": {}, "displayType": "dropdownlist", "shuffleChoices": true, "unitTests": [], "maxMarks": 0, "customMarkingAlgorithm": "", "type": "1_n_2"}], "type": "gapfill", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Hence, find the general solution of the differential equation
\n$\\Big( \\simplify{{aa}*{bb}} x^{\\simplify{{bb}-1}} y^{\\var{{cc}}} + \\var{{dd}} \\cos (y) \\Big) \\mathrm{d} x + \\Big( \\simplify{{aa}*{cc}} x^{\\var{{bb}}} y^{\\simplify{{cc}-1}} - (\\var{{dd}} x + \\var{{ff}}) \\sin (y) + \\cos(y) \\Big) \\mathrm{d} y$.
\nYou may find it helpful to look at section 38.4 of the lecture notes, particularly example 38.4.1 . The solution will be in the form $F(x,y) = C$ for some arbitrary constant $C$. You need not attempt to find an explicit solution; that is, you need not try to find a solution in the form $y(x) = \\ldots$. Also, if possible, you should absorb all constants into the constant C.
\nThe solution is
\n[[0]] $= C$.
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkVariableNames": false, "answer": "{aa}*(x^({bb}))*(y^({cc})) + ({dd}*x + {ff})*cos(y) + sin(y)", "unitTests": [], "expectedVariableNames": [], "checkingType": "absdiff", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "type": "jme", "customMarkingAlgorithm": "", "marks": "5"}], "type": "gapfill", "customMarkingAlgorithm": "", "unitTests": []}], "statement": "Lent Term Week 9 (lectures 37 and 38): In this question you will look at linear ODEs with constant coefficients, and exact ODEs.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
This is the question for Lent Term week 9 of the MA100 course at the LSE. It looks at material from chapters 37 and 38.
"}, "advice": "", "variables": {"B_coeff_1": {"definition": "g^2 + h +2*g*f", "templateType": "anything", "name": "B_coeff_1", "group": "Variables for part b", "description": "This is the coefficient of m^1 in our auxiliary equation.
"}, "B_root_3": {"definition": "g - (sqrt(h))*i", "templateType": "anything", "name": "B_root_3", "group": "Variables for part b", "description": "This is the third root of our auxiliary equation.
"}, "aa": {"definition": "random(3..7)", "templateType": "anything", "name": "aa", "group": "Variables for parts c and d", "description": "This is used in the definition of our exact ODE.
"}, "b": {"definition": "random(2..7 except(a))", "templateType": "anything", "name": "b", "group": "Variables for part a", "description": "This is the second root of the auxiliary equation. It is a repeated root.
"}, "A_coeff_4": {"definition": "-2*c -2*b - a", "templateType": "anything", "name": "A_coeff_4", "group": "Variables for part a", "description": "This is the coefficient of m^4 in the auxilliary equation.
"}, "d": {"definition": "random(2,3,5,6,7)", "templateType": "anything", "name": "d", "group": "Variables for part a", "description": "c+sqrt(d)*i is the third root of the auxiliary equation, andc-sqrt(d)*i is thefourthroot of the auxiliary equation.
"}, "A_coeff_5": {"definition": "1", "templateType": "anything", "name": "A_coeff_5", "group": "Variables for part a", "description": "This is the coefficient of m^5 in the auxilliary equation.
"}, "B_coeff_0": {"definition": "-(g^2)*f - h*f", "templateType": "anything", "name": "B_coeff_0", "group": "Variables for part b", "description": "This is the coefficient of m^0 in our auxiliary equation.
"}, "bb": {"definition": "random(3..7)", "templateType": "anything", "name": "bb", "group": "Variables for parts c and d", "description": "This is used in the definition of our exact ODE.
"}, "B_PI_3": {"definition": "1/(1 - 2*g - f + g^2 + h + 2*g*f - (g^2)*f - h*f)", "templateType": "anything", "name": "B_PI_3", "group": "Variables for part b", "description": "This is the coefficient of e^x in our particular integral.
"}, "B_PI_1": {"definition": "-1/((g^2)*f + h*f)", "templateType": "anything", "name": "B_PI_1", "group": "Variables for part b", "description": "This is the coefficient of x in our particular integral.
"}, "c": {"definition": "random(2..8 except(a) except(b))", "templateType": "anything", "name": "c", "group": "Variables for part a", "description": "c+sqrt(d)*i is the third root of the auxiliary equation, and c-sqrt(d)*i is the fourth root of the auxiliary equation.
"}, "B_root_1": {"definition": "f", "templateType": "anything", "name": "B_root_1", "group": "Variables for part b", "description": "This is the first root of our auxiliary equation.
"}, "a": {"definition": "random(2..6)", "templateType": "anything", "name": "a", "group": "Variables for part a", "description": "This is the first root of the auxiliary equation.
"}, "B_root_2": {"definition": "g + (sqrt(h))*i", "templateType": "anything", "name": "B_root_2", "group": "Variables for part b", "description": "This is the second root of our auxiliary equation.
"}, "A_coeff_3": {"definition": "b^2 + c^2 + d + 2*a*b + 4*b*c + 2*a*c", "templateType": "anything", "name": "A_coeff_3", "group": "Variables for part a", "description": "This is the coefficient of m^3 in the auxilliary equation.
"}, "dd": {"definition": "random(3..7)", "templateType": "anything", "name": "dd", "group": "Variables for parts c and d", "description": "This is used in the definition of our exact ODE.
"}, "A_coeff_0": {"definition": "-a*(b^2)*(c^2) - a*(b^2)*d", "templateType": "anything", "name": "A_coeff_0", "group": "Variables for part a", "description": "This is the coefficient of m^0 in the auxilliary equation.
"}, "cc": {"definition": "random(3..7)", "templateType": "anything", "name": "cc", "group": "Variables for parts c and d", "description": "This is used in the definition of our exact ODE.
"}, "h": {"definition": "random(2..7 except(4))", "templateType": "anything", "name": "h", "group": "Variables for part b", "description": "g + sqrt(h)*i is the second root of our auxiliary equation, and g - sqrt(h)*i is the third.
"}, "B_PI_2": {"definition": "-(g^2 + h +2*g*f)/(((g^2)*f + h*f)^2)", "templateType": "anything", "name": "B_PI_2", "group": "Variables for part b", "description": "This is the constant coefficient in our particular integral.
"}, "ff": {"definition": "random(3..7)", "templateType": "anything", "name": "ff", "group": "Variables for parts c and d", "description": "This is used in the definition of our exact ODE.
"}, "A_Coeff_1": {"definition": "2*a*(b^2)*c + 2*a*b*c^2 + (b^2)*(c^2) + (b^2)*d + 2*a*b*d", "templateType": "anything", "name": "A_Coeff_1", "group": "Variables for part a", "description": "This is the coefficient of m^1 in the auxilliary equation.
"}, "g": {"definition": "random(2..6)", "templateType": "anything", "name": "g", "group": "Variables for part b", "description": "g + sqrt(h)*i is the second root of our auxiliary equation, and g - sqrt(h)*i is the third.
"}, "f": {"definition": "random(2..6)", "templateType": "anything", "name": "f", "group": "Variables for part b", "description": "This is the first root of our auxiliary equation.
"}, "B_coeff_3": {"definition": "1", "templateType": "anything", "name": "B_coeff_3", "group": "Variables for part b", "description": "This is the coefficient of m^3 in our auxiliary equation.
"}, "A_coeff_2": {"definition": "-(a*b^2 + a*c^2 + 2*b*c^2 + 2*c*b^2 + 4*a*b*c + 2*b*d + a*d)", "templateType": "anything", "name": "A_coeff_2", "group": "Variables for part a", "description": "This is the coefficient of m^2 in the auxilliary equation.
"}, "B_coeff_2": {"definition": "-2*g - f", "templateType": "anything", "name": "B_coeff_2", "group": "Variables for part b", "description": "This is the coefficient of m^2 in our auxiliary equation.
"}}, "ungrouped_variables": [], "extensions": [], "functions": {}, "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}