Question

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 \(x>0\).

Answer 1

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 \(C_1 \frac{\sin x}{\sqrt{x}}+C_2 \frac{\cos x}{\sqrt{x}}=\frac{C_1 \sin x+C_2 \cos x}{\sqrt{x}}\).