We investigated the composition dependence of the finite extensible nonlinear elastic (FENE) constant of poly(ethylene oxide) (PEO) solutions by the Tikhonov regularization method with the measured normal stress in microfluidic channels. We measured the normal stress of the PEO solutions in response to the extensional flows in microfluidic channels. Additionally, we extracted the extensional relaxation time distributions and the FENE constants of the PEO solutions from the measured normal stress by inverse calculations. The linear average value of the extensional relaxation times was consistent with that measured using a capillary breakup extensional rheometer. The shortest extensional relaxation time was close to the mean value of the Zimm relaxation time distribution. The FENE constant evaluated in the semidilute regime was 10.3 times higher than that in the dilute regime. The proposed method is promising for the determination of the Zimm time, FENE constant, as well as the extensional relaxation times of low-viscosity solutions.