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

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?

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.

• 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) – Michael E2 Jan 15 at 12:50