Question

DIFFERENTIAL EQUATIONS:
(x^2+3*x^3) y '' + x y' + (x^2 - 5) y = 0
FIND THE RECURSION FORMULA FOR THE COEFFICIENTS AT Xo=0 FOR THE LARGER ROOT OF THE INDICIAL EQUATION. COMPUTE a1, a2, a3 when ao=1.

Answer 1

\(y = \sum_{n=0}^\infty a_n x^n\\(x^2 -5)y = \sum_{n=0}^\infty a_n x^{n+2}-5a_n x^n\)

\(y'=\sum_{n=1}^\infty na_n x^{n-1}\\xy'=\sum_{n=1}^\infty na_n x^n\)

\(y"=\sum_{n=2}^\infty n(n-1)a_n x^{n-2}\\(x^2 +3x^3)y" =\sum_{n=2}^\infty n(n-1)a_n x^n+3 n(n-1)a_n x^{n+1}\)

Combine all and simplify more, you can get the answer for recursive formula.