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...Using denite integrals to solve a separable initial value problem louis talman ph d emeritus professor of mathematics metropolitan state university denver february the plan given dierential equation dy f x g y with condition dx when we begin by supposing that gives solution wenote if then constant function is thus may suppose doesn t vanish near so at least close enough choose have z dt substitution theorem for allows us make u on left side this du and fact see can now rewrite as xf know functions which write or it remains only in terms finishing depends course our ability carry out whatever algebra needed invert an example consider yx e use standard formal calculation correct form though perhaps not clearly content avoid surfeit notation explicit first becomes also because are going want upper limits be variables but s unwise one symbol two dierent meanings same expression turn crank ln exp where used expu point recall account intitial must negative values such fraction positive there...