I am aware that there are books treating calculus solely from the infinitesimal perpective. Are there books doing the same for ordinary/partial differential equations? I would be interested in introductory to advanced books - if such books exist.
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Nonstandard analysis, a practical guide with applications by Lutz and Goze, (Springer Lecture Notes in Mathematics #881) has a large section (150 pages) on perturbation analysis.
Nonstandard analysis by Robert (Dover publications) has about 15 pages on DEs and Green functions.
...but I don't know any text completely devoted to nonstandard methods for DEs.
Keisler's Foundations of Infinitesimal Calculus has a chapter (chapter 14) on differential equations, at http://www.math.wisc.edu/~keisler/foundations.html