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Table.pdf - MATH 241 - Partial Differential Equations (Exam...

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MATH 241-Partial Differential Equations(Exam#3)Summer Semester, 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 thaty1xis one non-zero solution to the second-order, linear, homogeneous ODE,a2xy′′xa1xyxa0xyx0fora2x

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