APPM1026A-MMMS2-Quiz 1
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APPM 1026A
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Dec 30, 2024
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Dashboard/My courses/APPM1026A-MMM1-S2-2024/Quizzes/Quiz 1Started onFriday, 9 August 2024, 1:41 AMStateFinishedCompleted onFriday, 9 August 2024, 1:46 AMTime taken4 mins 40 secsGrade6.00out of 10.00 (60%)Question 1CorrectMark 1.00 out of 1.00Which of the following is not a criterion for classifying an ordinary differential equation by linearity?A.No product terms of the dependent variable and/or any of its derivatives are presentB.The differential equation is homogeneousCorrect!Please see general feedback.C.The dependent variable and its derivatives are of the first degreeD.No transcendental functions of the dependent variable and/or its derivatives occurYour answer is correct.Alineardifferential equation is one in which neither the unknown function nor any of its derivatives are raised to a power greaterthan 1.An nth -order ODE is said to be linear ifthe dependent variable andallof its derivatives are of the first degree, that is, the power of each term involving the dependentvariable is 1,andthe coefficients of the dependent variable and any of its derivatives are constants,orthey depend on the independentvariable only.A differential equation that is not linear is said to benon-linearand this occurs when the above does not hold, that is, non-linearfunctions of the dependent variable or any of its derivatives occur in the differential equation.The correct answer is:The differential equation is homogeneous
Question 2IncorrectMark 0.00 out of 1.00Question 3CorrectMark 1.00 out of 1.00Which oneof the following is not a second-order differential equation?Select one:A., whereB.This equation is second order as the highest derivative is the second derivative in Leibniznotation .C., whereD., where+ 3+ sint= 0x¨x˙x=x(t)t−2+x= 0dxdt()xd2dt24xd2dt2+ 2= 3 +y′′x2y(4)y=y(x)+= 0y′′y′yy=y(x)This question is testing your understanding of classifying a DE by order. The equationis not a second-orderDE as the highest derivative is not the second derivative. The highest derivative is the fourth derivative in prime notation, and hencethe DE is fourth order. Note that the prime notationdepicts the fourth derivative, NOT.The correct answer is: , where+ 2= 3 +y′′x2y(4)y(4)y4+ 2= 3 +y′′x2y(4)y=y(x)Consider the differential equation:,where .With respect to the dependent variable, the differential equation isSelect one:A.homogeneous as the differential equation is linearB.homogeneous as the dependent variable and all its derivatives are of degree 1C.non-homogenous due to the term Correct. See the General Feedback.D.non-homogeneous as the differential equation is non-linear(1−x)−4x+ 5y= cosxy′′y′y=y(x)cosxThis question tests your understanding of the two definitions of homogeneity, and knowing when to implement which definition.Since the DEis linear, we only need to look for the occurrence of a term involving the independentvariable only (including a constant term/non-zero function of the independent variable). Under this definition, the DE is non-homogenous due to the term , i.e..The correct answer is: non-homogenous due to the term (1−x)−4x+ 5y= cosxy′′y′cosxg(x)≠0cosx
Question 4IncorrectMark 0.00 out of 2.00Question 5CorrectMark 1.00 out of 1.00Which oneof the following differential equations is fourth order, linear?Select one:A.B., whereC., whereD.This equation is fourth order, but is is non-linear as it contains a product of the dependent variablewith its first derivative, i.e..−+y= 0()yd2dx22dydx−2y=xy′′y(4)y=y(x)(−2y= 4xy′)4y=y(x)+y= 0yd4dx4dydxyy′This question is checking your holistic understanding of classification of ODEs, particularly around order and linearity. Theequationis fourth order, linear. it does not contain powers of more than one of the dependent variable or itsderivatives. It also does not contain products of the dependent variable with itself or any of its derivatives, nor does it contain anyother non-linear functions of the dependent variable. It is fourth-order as the highest derivative is the fourth derivative in primenotation.The correct answer is: , where−2y=xy′′y(4)−2y=xy′′y(4)y=y(x)Which oneof the following differential equations is non-linear?Select one:A., whereB., whereC., whereD.Correct. See general feedback.+= 5xx¨etx=x(t)++ 2= 3y(5)y′′x2y=y(x)(1−2x)+ 2x+ 4y= cosxy′′y′y=y(x)−= 0yd2dx21ydydxThis question is testing your understanding of the linearity of a differential equation. Based on the definition of linearity, it is clearthat the equationis non-linear due to the term. This constitutes a product of a function of the dependentvariable with one of its derivatives.The correct answer is: −= 0yd2dx21ydydx1y−= 0yd2dx21ydydx
Question 6CorrectMark 2.00 out of 2.00Question 7IncorrectMark 0.00 out of 1.00Consider the differential equation:.The differential equation isA.linear, as there are no non-linear functions ofthe independent variableAny non-linear functions in the independent variable do not affect thelinearity of a DE. See general feedback.B.linear, as the differential equation ishomogeneousThis differential equation is non-linear. Please recheck the definition of linearity,and then the general feedback.C.non-linear, due tothe term This answer is partially correct. However it does not account for the fact that the secondderivative is multiplied by the dependent variable.D.non-linear,due to the This answer is partially correct. However it does not account for the fact that the equation contains anon-linear term that is a function of the dependent variable, .y−=yd2dx2dydxeyeyyeyThis question tests whether you understand what parts of an equation dictates its linearity/non-linearity. The above DE is non-linear1. due to the term , as it depicts a product of the dependent variable with its second derivative, and2. the term, which is a non-linear function of the dependent variable.The correct answers are: non-linear, due to the term , non-linear, due to the yeyeyyChoose the mostcorrect answer.A differential equation is considered to be ordinary if it hasA.one dependent variableB.more than one dependent variableC.more than one independent variableD.one independent variableE.none of these optionsYour answer is incorrect.Anordinary differential equation (ODE)is an equation that involves someordinaryderivatives of one or more unknown function(the dependent variable(s)) ofone single independent variable.The correct answer is:one independent variable
Question 8CorrectMark 1.00 out of 1.00Which oneof the following differential equations is homogeneous?Select one:A., whereB., whereCorrect. Check the General Feedback.C.D.y+x= 0y′′y=y(x)+x=yx2y′′y′x2y=y(x)+ 4y−2 = 0dydx+ 3−4+ 6x= 0yd3dx3yd2dx2dydxThis question relies on your understanding of the definition of homogeneity, both in terms of the test for homogeneity for all classesof DE's, as well as the refined definition for linear DE's. In light of this, the equationis homogenous:1. by undergoing the test for homogeneity which shows it is invariant under a scalar transformation of the dependent variable,and2. since it is linear, it is homogenous as it does not contain a single term involving the independent variable only, i.e..The correct answer is: , where+x=yx2y′′y′x2g(x) = 0+x=yx2y′′y′x2y=y(x)◀Test 2 - ResultsJump to...Quiz 2 ▶