y"+(x^2)*y'+x*y=0

Question

how to solve it stepby step.

Answer 1

Let y=∑a_nxⁿ for n≥0.

The first few terms are therefore:

y=a₀+a₁x+a₂x²+a₃x³+... (general term a_nxⁿ),

y'=a₁+2a₂x+3a₃x²+... (general term na_nx^n-1),

y"=2a₂+6a₃x+12a₄x²+20a₅x³... (general term n(n-1)a_nx^n-2),

x²y'=a₁x²+2a₂x³+3a₃x⁴+... (general term na_nxⁿ⁺¹),

xy=a₀x+a₁x²+a₂x³+a₃x⁴+... (general term a_nxⁿ⁺¹).

Add the last three equations together:

2a₂+(a₀+6a₃)x+(2a₁+12a₄)x²+(3a₂+20a₅)x³+... 

(general term (na_n-1+(n+1)(n+2)a_n+2)xⁿ)=0

Each coefficient of x must be zero for the whole series to sum to zero for general x.

Therefore a₂=0. a₀ and a₁ are unknown constants. Let a₀=A and a₁=B.

6a₃=-A, a₃=-A/6; a₄=-B/6; a₅=0; a_n+2=-na_n-1/[(n+1)(n+2)].

This makes y=A+Bx-Ax³/6-Bx⁴/6+Ax⁶/45+5Bx⁷/252-...

Note that the coefficient for x², x⁵, x⁸, etc., are zero.