How to find Mclaurins series which satisfy differential equations?

Pretty much plug and chug:

McLaurin series: y(x) = ∑ₙ (aₙ xⁿ), summed on "n" from zero to infinity.

    Substituting x=0 in y(0)=1 gives a₀=1.

Then, computing each term in the given equation:

    y'(x) = ∑ₙ ((n+1)aₙ₊₁ xⁿ)

    x×y(x) = ∑ₙ (aₙ₋₁ xⁿ)

    y(x)² = ∑ₙ ((∑ₖ aₖaₙ₋ₖ) xⁿ), summed on "k" from 0 to "n".

    1 = ∑ₙ (δ₀ₙ xⁿ), where δ₀₀=1 and, for n≠0, δ₀ₙ=0.

Shows the given equation as:

    ∑ₙ ((n+1)aₙ₊₁ xⁿ) = ∑ₙ (((∑ₖ aₖaₙ₋ₖ) + aₙ₋₁ + δ₀ₙ) xⁿ)

Equate like powers of x produces the recursion:

    aₙ₊₁ = ((∑ₖ aₖaₙ₋ₖ) + aₙ₋₁ + δ₀ₙ) / (n+1), for n≥0 {and a₋₁=0}

Substituting n=0 and a₀=1 gives a₁=2, then substituting that gives a₂=2.5, then ...

So the required McLaurin series looks like:

    y(x) = 1 + 2x + 2½x² + 3⅔x³ + ...

Not a function I recognize...