solve the following differential equation

Question

x² d²y/dx² - 7x dy/dx + 16y = (ln x + 1)x^4

Answer 1

Let y=∑a_nxⁿ, then y'=∑na_nx^n-1, y"=∑n(n-1)a_nx^n-2 for integer n≥0.

16y=16a₀+16a₁x+16a₂x²+16a₃x³+16a₄x⁴+16a₅x⁵+...

-7xy'=-7a₁x-14a₂x²-21a₃x³-28a₄x⁴-35a₅x⁵-...

x²y"=2a₂x²+6a₃x³+12a₄x⁴+20a₅x⁵+...

The DE LHS becomes:

16a₀+9a₁x+4a₂x²+a₃x³+∑(n-4)²a_nxⁿ for n≥4 (x⁴ has a coefficient of 0).

The DE RHS expands to:

x⁴(x-x²/2+x³/3-x⁴/4+...) so the lowest power of x is x⁵. Therefore a₀ to a₄ are zero.

x⁵ has a coefficient of 1, x⁶ a coefficient of 1/2, x⁷ a coefficient of 1/3, xⁿ a coefficient of 1/(n-4).

Equating LHS and RHS coefficients:

(n-4)²a_n=1/(n-4), from which a_n=1/(n-4)³.

y=∑xⁿ/(n-4)³ for n≥5, y=x⁵+x⁶/8+x⁷/27+x⁸/64+...