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Here are three implicit function equations

AsymptoticSolve[(a*y^2 + Sin[x])^2 == x, y, {x, 1, 3}]
AsymptoticSolve[(a*Sin[y]^2 + Sin[x])^2 == x, y, {x, 1, 3}]
AsymptoticSolve[(a*y^2 + Sin[x*y])^2 == x, y, {x, 1, 3}]

Why can we only find the asymptotic expression of the solution of the first implicit function?

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The AsymptoticSolve command works with the result of the Solve command. Let us consider these results.

  1. Solve[(a*Sin[y]^2 + Sin[x])^2 == x, y] results in 8 expressions of the form y->ConditionalExpression which are infinitely valued because of terms 2 \[Pi] C[1], C[1] \[Element] Integers . I think this circumstance makes difficulties. The following works as a workaround.

    Table[Series[y /. Solve[(a*Sin[y]^2 + Sin[x])^2 ==x,y][[j]],{x, 1, 3}],{j, 1, 8}]

  2. No comment in view of

    Solve[(a*y^2 + Sin[x*y])^2 == x, y]

Solve::nsmet: This system cannot be solved with the methods available to Solve.

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  • 2
    $\begingroup$ For (2), one may also pick a particular solution as an initial condition for y: for example, AsymptoticSolve[(a*Sin[y]^2 + Sin[x])^2 == x, {y, 1/2 ArcCos[(-2 + a + 2 Sin[1])/a]}, {x, 1, 3}] (+1) $\endgroup$ – Michael E2 Jan 15 at 12:50

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