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Is there a method for solving the following system of generalized Abel's integral equation:?

$(x^2 -1)\int_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt =g_1 (x),\\ x^3 \int_0^x \frac{u(t)}{(x-t)^{\frac{1}{4}}}\; dt + (1-x)\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{5}}}\; dt =g_2 (x),$

where

$\begin{cases} ‎g_1 (x)&=‎\frac{16}{15}x^{‎\frac{9}{2}‎}-‎\frac{16}{15}x^{‎\frac{5}{2}‎}+‎\frac{27}{40}x^{‎\frac{11}{3}‎}+‎\frac{243}{440}x^{‎\frac{14}{3}‎}, ‎\\ ‎g_2 (x)&= ‎\frac{128}{231}‎x^{‎\frac{23}{4}‎}+‎\frac{125}{252}x^{‎\frac{14}{5}‎}-‎\frac{125}{1197}x^{‎\frac{19}{5}‎}-‎\frac{625}{1596}x^{‎\frac{24}{5}‎}\end{cases}$
with $0\leq x\leq 1$?

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The system in question is solved this article: A new operational method to solve Abel’s and generalized Abel’s integral equations.see here

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