\[\large u = \csc^{-1}\large \sqrt{\dfrac{\sqrt{x}+ \sqrt{y}}{\sqrt[3]{x} + \sqrt[3]{y}}}\]

\[\large x^2\dfrac{\partial^{2}u}{\partial x^{2}} + 2xy\dfrac{\partial^{2}u }{\partial x\partial y} +y^{2}\dfrac{\partial^{2} u}{\partial y^{2}} =\dfrac{\tan{u}}{a^{2}}\left[b+ \tan^2{u} \right]\]

Find \(a+b\), where \(a\) and \(b\) are coprime positive integers.

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