MATH 241-Partial Differential Equations(Homework#2)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(20 Points) -Constructing Possible Solutions To a PDESuppose thatxandyareindependentvariables and that∂ux,y∂xx∂ux,y∂yfor 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′′xx′xx3for 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,′′x2′xxe2xcosxfor−x .Hint: When solving for a particular solution to this ODE, try guessing asolution of the form

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———————————————————————————————————————Problem#5(20 points) -Computing A General Solution To Linear PDEBCsDetermine ageneral solutionto the heat equation———————————————————————————————————————

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