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Exam2_s.pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Exam#2)Summer Semester I, 2021M. Carchidi———————————————————————————————————————Solving The ODE:ay′′xbyxcyx0 (a,b,cconstants witha0)Db24acy1xy2xNegativeexcosxexsinxb/2aD/2aZeroexxexb/2aPositiveexcoshxexsinhxb/2aD/2aexexb/2aD/2ayxc1y1xc2y2x———————————————————————————————————————Given the ODEn′′xnnx0for 0xL, we have, for the four basic boundary conditions (BCs), the following table ofresults.CaseBCsnnnxnn1n0nL01,2,3,...n/LsinxnL/22n0nL01,2,3,...n1/2/LcosxnL/23n0nL01,2,3,...n1/2/LsinxnL/24an0nL01,2,3,...n/LcosxnL/24bAlson0000x1Lwherefg0Lfxgxdxandpq0whenpq.———————————————————————————————————————
———————————————————————————————————————Given the PDEa2ux,tx2b2ux,txtc2ux,tt20witha,bandcconstants, the following table summarizes a general solution for these.Solving The PDE:auxxx,tbuxtx,tcuttx,t0 (a,b,cconstants)Case #ConditionsQuadratic EquationRootsux,t1c0b24ac0abrcr20r,rfxrtgxrt2c0b24ac0abrcr20r0,r0fxr0ttgxr0t3a0b24ac0as2bsc0s,sfsxtgsxt4a0b24ac0as2bsc0s0,s0fs0xtxgs0xt5ac0b0fxgt———————————————————————————————————————2
———————————————————————————————————————Problem#1(25 Points) -Solving a PDE With Two ICs and One BCDetermine theuniquesolution to the following PDE———————————————————————————————————————
Problem#2(25 Points) -A General Solution to a Wave-Like PDE and BCsConstruct ageneral solutionto the wave equation2ux,tx22ux,tt22for 0x1 and 0t, given the boundary conditionsu0,t0andux,txx11for all 0t.Hint: You may use the above table for the eigenvalue/eigenfunction probleminvolving the ODEn′′xnnx0 in order to save some time.

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Boundary value problem

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