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ENM510 - Advanced Engineering Mathematics I (Final Exam)Fall Semester, 2016M. Carchidi––––––––––––––––––––––––––––––––––––Useful Information: Some Special Regular Sturm-Liouville ProblemsThe eigenfunctions and corresponding eigenvalues to the ODEϕ00n(x) +λ2nϕn(x) = 0for0< x < L, subject to the boundary conditions (BCs) given in the tablebelow are given by theλn’s andϕn(x)’s as summarized in the followingtable.CountBC atx= 0BC atx=Lλn(n1)ϕn(x)(n1)λ0ϕ0(x)1ϕ(0) = 0ϕ(L) = 0nπ/Lsin(λnx)2ϕ0(0) = 0ϕ(L) = 0(n1/2)π/Lcos(λnx)3ϕ(0) = 0ϕ0(L) = 0(n1/2)π/Lsin(λnx)4ϕ0(0) = 0ϕ0(L) = 0nπ/Lcos(λnx)01It should also be noted that in all four of these casesZL0ϕm(x)ϕn(x)dx=0,whenm6=nL/2,whenm=n6= 0and (in case 4)ZL0ϕn(x)dx=0,whenn6= 0L,whenn= 0sinceϕ0(x) = 1for case 4 above.––––––––––––––––––––––––––––––––––––
––––––––––––––––––––––––––––––––––––Problem #1 (25 points)Determine the functionymin(x)in the subspace defined byW={y(x)|xy00(x)(x+ 1)y0(x) +y(x) = 0forx >0},that isclosestto the functiony0(x) = 1, with respect to the inner producthf, gi=Z0f(x)g(x)e3xdxdefined inW.Hint: You may use the fact thatZxeaxdx=(ax1)a2eaxandZ0xneaxdx=n!an+1fora >0andn= 0,1,2, ..., along with"abcd#1=1adbc"dbca#foradbc6= 0.
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––––––––––––––––––––––––––––––––––––Problem #2 (25 points)Determine acompletesolution foru(x, t)in the region
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––––––––––––––––––––––––––––––––––––Problem #3 (25 points)Solve formally foru(x, y)in the rectangular regionR={(x, y)|0x1,0y1},given the partial differential equation (PDE)2u(x, y)x2+2u(x, y)y2= 0,for0< x <1and0< y <1, the boundary conditions (BCs)u(x, y)x¯¯¯¯¯x=0=f(y)andu(1, y) = 0,for0y1, and the boundary conditions (BCs)u(x,0) =g(x)andu(x,1) = 0,for0x1.––––––––––––––––––––––––––––––––––––4
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