# hmk#9_s (1).pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Homework#9)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(25 Points) -A Regular Sturm-Liouville Problem(RSLP)a.) (20 Points) Determine the eigenvaluesand corresponding non-zero eigenfunctionsxto the RSLP consisting of the ODE′′xx0for 0x1 with the BCs000 and10, and be sure to check0,0 and0.b.) (5 Points) Make plots of1x,2xand3x, showing that when then’s are orderedas123,the eigenfunctionnxgoes through zero exactlyn1 times in the open interval0x1.———————————————————————————————————————Problem#2(25 Points) -Another Regular Sturm-Liouville Problem(RSLP)a.) (15 Points) Determine the eigenvaluesand corresponding non-zero eigenfunctionsxto the RSLP consisting of the ODEx2′′x3xxx8x0for 0x1 with the BCs00 and10.Hint: See Problem #5 of Homeworkb.) (5 Points) Construct a "dot product" for which0 whenmnand also compute
mnnn.c.) (5 Points) If a piecewise continuous functionfxin the interval 0x1 is expressed asfxnannxfor 0x1, determine an expression for theanin terms offxandnx.1———————————————————————————————————————
———————————————————————————————————————Problem#3(25 points) -Another Regular Sturm-Liouville Problem(RSLP)a.) (15 Points) Determine the eigenvaluesand corresponding non-zero eigenfunctionsxto the RSLP consisting of the ODE′′xxe2xx0for 0xwith the BCs00 and0.Hint: See Problem #5 of Homeworkb.) (5 Points) Make plots ofx,xandx, showing that when the’s are ordered
123nas123,the eigenfunctionnxgoes through zero exactlyn1 times in the open interval0x.c.) (5 Points) If a piecewise continuous functionfxin the interval 0xis expressedasfxnannxfor 0x, determine an expression for theanin terms offxandnx.1———————————————————————————————————————Problem#4(25 points) -Another Regular Sturm-Liouville Problema.) (15 points) Determine the eigenvaluesand corresponding non-zero eigenfunctionsxto the RSLP consisting of the ODE′′x4xx4x22x0for 0x1 with the BCs00 and10.Hint: See Problem #4 of Homeworkb.) (5 Points) Make plots ofx,xandx, showing that when the’s are ordered
123nas123,the eigenfunctionnxgoes through zero exactlyn1 times in the open interval0x1.

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