Find the equation that doesn't have a series solution in form of $$y(x)=\sum_{n=0}^{\infty}a_nx^n $$

$$ 1.\;\;\frac{d^2y}{dx^2}+\frac{\sin x}{x}y=0 \qquad 2.\;\;\frac{d^2y}{dx^2}+\frac{\cos x}{x}y=0$$ $$ 3.\;\;\frac{d^2y}{dx^2}+{\sin(x)}y=0\qquad 4.\;\;\frac{d^2y}{dx^2}+{\cos(x)}y=0$$

$y''+P(x)y'+Q(x)y=0 $

It seems $Q(x)$ in every choice is analytic to me. How should I solve it?

  • 3
    $\begingroup$ whats the limit $\cos(x)/x$ as $x\to 0$? $\endgroup$ Jul 27 '18 at 12:28
  • $\begingroup$ In one of these choices, $Q(x)$ is not analytic at $x=0$ $\endgroup$
    – Dylan
    Jul 27 '18 at 12:45
  • $\begingroup$ Isn't cosx/x representable as power series which means analytic? $\endgroup$
    – NK Yu
    Jul 27 '18 at 12:52
  • 1
    $\begingroup$ Try to find the power series of $\cos x/x$. You'll see that the leading term is $1/x$ $\endgroup$
    – Dylan
    Jul 27 '18 at 12:53
  • 1
    $\begingroup$ @NKYu Do the same calculation for $\frac {\sin x }x$ and see the difference. $\endgroup$
    – MtGlasser
    Jul 27 '18 at 13:03

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