The elastic scattering and one-pion-production process in proton-proton collision at 0.97 GeV are analyzed in terms of the one-boson-exchange model considering the effects of the form factors for baryon-meson interactions. The pion production is assumed to proceed through Δ(1236) resonance production. The one-π, one-ρ, one-ω and one-scalar-meson exchanges are taken for the elastic proton-proton scattering amplitudes TA. Similarly, the one-π and one-ρ exchanges are taken for the Δ(1236)-resonance production amplitudes TB. In order to satisfy the unitarity requirements the K-matrix method is used. The observed energy distribution of recoil neutron in the one-pion-production process as well as the experimental data on elastic proton-proton scattering can be reproduced well by this model. The ρ contribution to TB is found to be very small. This is quite different from the conclusion of our previous attempt. Using the values of parameters obtained by analyzing the data at 0.97 GeV, experimental data on both the elastic and inelastic processes at 1.4 GeV and 2 GeV are fairly well explained.