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I'm really having a hard time finding a book that covers distributional equations such as $xT = \delta(x)$ or distributional differential equations. Any suggestions?

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This is the problem of division of distributions. The general existence theorem is the Malgrange-Ehrenpreis Theorem. The first two volumes of Hörmander's "The Analysis of Linear Partial Differential Operators" is a canonical reference. See also this book by Ortner and Wagner.

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In addition to @AbdelmalekAbdesselam's mention of the canonical Hormander, there are some "lighter" physical-book sources that also illustrate operational use of the ideas. The best conventional book I know is G. Grubb's "Distributions and operators". (She is in Hormander's "school"...)

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