Exam1_s.pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Exam#1)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(25 Points) -An Eigenvalue/Eigenfunction ProblemDetermine all real scalarsandnon-zerosolutionsx, given the ODE———————————————————————————————————————
Problem#2(25 Points) -A Time-Independent Solution to a PDE and BCsConstruct the time-independent solution to the PDE
Problem#3(30 points) -A Solution to a PDE and BCsDetermine the complete solution to the heat equation———————————————————————————————————————
Problem#4(20 points) -Working With A Dot ProductSuppose that the dot (inner) product between two functionsfandgis defined byfg0fxgxdxand suppose thatfxexandgxe2x. With respect to this dot product,a.) (7 points) determine the "length" of each of these functions,b.) (6 points) determine the "angle" between these function, andb.) (7 points) determine the values ofAandB, if the functionhxAe3xBe4xexis tobe "perpendicular" to both the givenfandg.Hint: You may use the fact that0eaxdx1afora0.———————————————————————————————————————
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