Course Hero Logo

Su20_Exam1_s.pdf - MATH 241 - Partial Differential...

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 1 - 3 out of 9 pages.

MATH 241-Partial Differential Equations(Exam#1)Summer Semester I, 2020M. Carchidi———————————————————————————————————————A general solution to the first-order, linear, non-homogeneous ODEa1xyxa0xyxbxfora1x0 is given byyx1xxbxa1xdxcwherexexpa0xa1xdxandcis an arbitrary constant.———————————————————————————————————————Solving The ODE:ay′′xbyxcyx0 (a,b,cconstants witha0)Db24acy1xy2xNegativeexcosxexsinxb/2aD/2aZeroexxexb/2aPositiveexcoshxexsinhxb/2aD/2aexexb/2aD/2ayxc1y1xc2y2x———————————————————————————————————————Solving The ODEax2y′′xbxyxcyx0 (a,b,cconstants witha0)Dba24acy1xy2xNegative|x|cosln|x||x|sinln|x|ba/2aD/2aZero|x|ln|x||x|ba/2aPositive|x|coshln|x||x|sinhln|x|ba/2aD/2a|x||x|ba/2aD/2ayxc1y1xc2y2x———————————————————————————————————————Given the second-order, linear, homogeneous ODE,a2xy′′xa1xyxa0xyx0fora2x0, ifa2xa1xa0x0, then one non-zero solution to this ODE isy1xex.———————————————————————————————————————
———————————————————————————————————————Given thaty1xis one non-zero solution to the second-order, linear, homogeneous ODE,a2xy′′xa1xyxa0xyx0fora2x0, a second linearly independent solution can be constructed usingy2xAy1xea1xa2xdxy12xdxfor any non-zero constantA.———————————————————————————————————————A general solution to the second-order, linear, non-homogeneous ODEa2xy′′xa1xyxa0xyxbxfora2x0 is given byyxyHxyPx, whereyHxc1y1xc2y2x, for arbitraryconstantsc1andc2, is a general solution to the above ODE whenbx0, andyPxxGt,xbta2tdtwithGt,xy1ty2xy1xy2ty1ty2ty1ty2t.———————————————————————————————————————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.

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 9 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Fall
Professor
N/A
Tags
Trigraph, D Generation X, x

Newly uploaded documents

Show More

Newly uploaded documents

Show More

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture