In this paper, a novel method is proposed to obtain an analytical solution for the non-stationary response of 1/2 order systems under stochastic excitation. The method first uses the eigenvector expansion to obtain the analytical expression of the impulse response function. Next, an analytical solution for the response power spectral density is obtained based on the Laplace transform. Through three illustrative numerical examples, including systems subjected to white noise, modulated white noise, and modulated colored noise with modified Kanai-Tajimi spectrum, the responses of the fractional systems are obtained analytically and compared to the pertinent Monte Carlo estimates to demonstrate the accuracy and applicability of the proposed method.