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Question: integro-diff. eq. reduction

Question:integro-diff. eq. reduction

Joseph Poveromo 25 Maple
ode differential-equations integro-differential-equation
September 17 2023
2

How could one reduce this equation int(x^2*diff(y(x), x)/(x^2 - 1), x) = int(y(x)^(1/2), x)^(-2/3)down to a first order ordinary differential equation? Maple can solve this equation, namely, 9*(1/x^(5/3) - 1/x^(11/3))*x*(sqrt(y(x))*x)^(8/3)/(8*(x^2 - 1)) - 3*x*(4*x^2 - 1)*(1/x^(5/3) - 1/x^(11/3))/(8*(x - 1)*(x + 1)) + _C1 = 0 , however I could not understand how this equation was arrived at, leading me to go to 'odeadvisor', which responded with y = G(x,y'(x)) labelled as the 'patterns' method, which appears to require a first order ode - that is why I raised the reduction question.
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