Obtain the differential equation by eliminating the arbitrary constants from \(y = c_1\cos(\log x) + c_2\sin(\log x)\).

Answer: \(x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0\)

\(y = c_1\cos(\log x) + c_2\sin(\log x)\) ... (1)

Differentiating w.r.t. \(x\): \(\frac{dy}{dx} = -c_1\sin(\log x)\cdot\frac{1}{x} + c_2\cos(\log x)\cdot\frac{1}{x}\)

\(\therefore x\frac{dy}{dx} = -c_1\sin(\log x) + c_2\cos(\log x)\)

Differentiating again: \(x\frac{d^2y}{dx^2} + \frac{dy}{dx} = -c_1\cos(\log x)\cdot\frac{1}{x} - c_2\sin(\log x)\cdot\frac{1}{x} = -\frac{y}{x}\) (using (1)).

Multiplying by \(x\): \(x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0\), which is the required differential equation.