# Form of series solution in Differential equation

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?

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