I wish to study the Abel equation and think it's solution involves several aspects of Calculus that others here may find interesting. I'll post the derivation of the solution in steps; that helps me learn it and I hope will provide incentive to others to comment about the process. I'm not really good with integral equations and welcome any comments; my understanding is quite lacking. I just like them and see the potential of them (and IDEs) being used in artificial intelligence.(adsbygoogle = window.adsbygoogle || []).push({});

Here's the equation:

[tex]f(x)=\int_0^x{\frac{\phi(y)}{\sqrt{x-y}}}[/tex]

With [itex] f(x)[/itex] given and [itex]\phi(x)[/itex] to be determined.

The first step in solving the equation is to define an integral operator, that is, an operator which takes a function and processes it with an integral. For this problem, Abel used the following integral operator:

[tex]I[f(x)]=\int_0^\xi{\frac{f(x)dx}{\sqrt{\xi-x}}}[/tex]

Thus, applying this operator to both sides of the equation yields:

[tex]\int_0^\xi{\frac{f(x)dx}{\sqrt{\xi-x}}}=\int_0^\xi{\frac{dx}{\sqrt{\xi-x}}}\int_0^x{\frac{\phi(y)}{\sqrt{x-y}}}dy[/tex]

The order of integration on the RHS needs to be changed next.

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# The Abel equation

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