I've lately came accross this "confusion" as I was solving a cauchy-euler equation that required a subtitution of for . It followed that Then or simply . But what about ? Is it or . I'm very confused about this because whenever we have a 1st order ODE let's say we change it to and then multiply the RHS by follows that as if the latter were a denominator!
If someone clears that out for me I would be very greatful!
[quote=Archie Meade;498895]Hi rebghb,
As is a ratio at any given x, being the slope of the tangent to the graph, you may treat as a denominator,
Thanks, well I am aware that is essentially a slope, but when we can treat as a denominator, can we treat as a numerator? (if the bottom part is the denominator then what about the upper part). I got your explanation using the definition of an integral (well not exactly definition) but if but what about ?? Why isn't it just the inverse as in . I'm lost because noone has made this clear, deriving has always been a routine