Let φ₁,φ₂ be independent solutions of $2x^2y\prime\prime-(x+x^2)y\prime+(x^2-2)y=0$, with φ₁(0) = 0. Find the smallest positive integer n for which $\lim_{x\to0}x^n\phi_2(x)/\phi_1(x)=0$.
The indicial equation at the regular singular point is $2r(r-1)-r-2=2r^2-3r-2=0$, with roots 2 and −1/2. The vanishing solution φ₁ has leading behaviour x², while an independent φ₂ has a nonzero x⁻¹ᐟ² component. Their ratio therefore scales as x⁻⁵ᐟ². Multiplication by xⁿ tends to zero exactly when n > 5/2. The smallest positive integer is 3, using the source’s local one-sided Frobenius convention.
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