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Solve $x^2 y^{\prime \prime}+x y^{\prime}+\left(x^2-\frac{1}{4}\right) y=0$ (a special case of Bessel's equation) for $x>0$.
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First check that $y_1=(\sin x) / \sqrt{x}$ is a solution for $x>0$.

Then, $v=\int \frac{x}{\sin ^2 x} \exp \left(-\int \frac{1}{x} d x\right) d x=$

$\int \frac{x}{\sin ^2 x} e^{-\ln x} d x$

$=\int \frac{x}{\sin ^2 x} \cdot \frac{1}{x} d x$

$=\int \csc ^2 x d x=-\cot x$.

Then $y_2=v y_1=$ $-\cot x \cdot \frac{\sin x}{\sqrt{x}}=-\frac{\cos x}{\sqrt{x}}$.

Hence, the general solution is $C_1 \frac{\sin x}{\sqrt{x}}+C_2 \frac{\cos x}{\sqrt{x}}=\frac{C_1 \sin x+C_2 \cos x}{\sqrt{x}}$.
by Platinum (141,884 points)

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