# Hmk#2_s.pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Homework#2)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(20 Points) -Constructing Possible Solutions To a PDESuppose thatxandyareindependentvariables and thatux,yxxux,yyfor allaxbandcyd. If non-zero solutions of the formux,yxyare tobe constructed for this PDE, then determine up to three arbitrary constants, the forms ofxandy.———————————————————————————————————————Problem#2(20 Points) -A First-Order,Linear,Non-Homogeneous ODEDetermine, up to one arbitrary constant, a general solution to the first-order, linear,non-homogeneous ODE,tdytdt2ytsintfor 0t.———————————————————————————————————————Problem#3(20 points) -A Second-Order,Linear,Non-Homogeneous ODEDetermine, up to two arbitrary constants, a general solution to the second-order, linear,non-homogeneous ODE,x2′′xxxx3for 0x.———————————————————————————————————————Problem#4(20 points) -A Second-Order,Linear,Non-Homogeneous ODEDetermine, up to two arbitrary constants, a general solution to the second-order, linear,non-homogeneous ODE,′′x2xxe2xcosxfor−x .Hint: When solving for a particular solution to this ODE, try guessing asolution of the form
———————————————————————————————————————Problem#5(20 points) -Computing A General Solution To Linear PDEBCsDetermine ageneral solutionto the heat equation———————————————————————————————————————
———————————————————————————————————————Solution to Problem#1Starting with the PDEux,yxxux,yywe setux,yxyand getxyxxywhich we write asxxxyyfor allaxbandcydwithxandyindependent. This can only happen whenxxxyyfor some constant

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Partial differential equation, x b