Base case: For \(n = 1\),

\(\frac{d}{dx}(xf(x)) = xf'(x) + f(x) = xf'(x) + (2(1)-1)f(x).\)

Assume true for \(n = k\), so

\(\frac{d^{2k-1}}{dx^{2k-1}}(xf(x)) = xf'(x) + (2k-1)f(x).\)

Then,

\(\frac{d^{2k}}{dx^{2k}}(xf(x)) = xf(x) + 2kf'(x).\)

\(\frac{d^{2k+1}}{dx^{2k+1}}(xf(x)) = xf'(x) + f(x) + 2kf(x)\)

\(= xf'(x) + (2k+1)f(x).\)

So, it is also true for \(n = k+1\). Hence, by induction, true for all positive integers.