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The Liouville-Green approximation givesHereThe control ratio $|q\prime |q^{-3/2}=O(x^{-1/2})$ tends to zero, verifying consistency.

Put $v=e^S$ and $p=S^{\prime }$. The Liouville-Green exponential ansatz turns the normal-form equation into the Riccati equationWriteMatching the constant, $x^{-1}$, and $x^{-2}$ coefficients gives

For the recessive branch,soSince $y=e^{x/2}{x}^{-1/2}v$, this givesThis agrees with the first two terms of part (a),Continuing the Riccati recursion reproduces the factorial asymptotic series, so $U$ is proportional to this recessive branch.

For the dominant branch,and in factsolves the phase equation exactly. It follows thatIndeed, $e^x$ is an exact second solution of the original differential equation.