Using a data sample corresponding to an integrated luminosity of 2.93 fb^{-1} recorded by the BESIII detector at a center-of-mass energy of 3.773 GeV, we present an analysis of the decays D^{0}→π^{-}π^{0}e^{+}ν_{e} and D^{+}→π^{-}π^{+}e^{+}ν_{e}. By performing a partial wave analysis, the π^{+}π^{-} S-wave contribution to D^{+}→π^{-}π^{+}e^{+}ν_{e} is observed to be (25.7±1.6±1.1)% with a statistical significance greater than 10σ, besides the dominant P-wave contribution. This is the first observation of the S-wave contribution. We measure the branching fractions B(D^{0}→ρ^{-}e^{+}ν_{e})=(1.445±0.058±0.039)×10^{-3}, B(D^{+}→ρ^{0}e^{+}ν_{e})=(1.860±0.070±0.061)×10^{-3}, and B(D^{+}→f_{0}(500)e^{+}ν_{e},f_{0}(500)→π^{+}π^{-})=(6.30±0.43±0.32)×10^{-4}. An upper limit of B(D^{+}→f_{0}(980)e^{+}ν_{e},f_{0}(980)→π^{+}π^{-})<2.8×10^{-5} is set at the 90% confidence level. We also obtain the hadronic form factor ratios of D→ρe^{+}ν_{e} at q^{2}=0 assuming the single-pole dominance parametrization: r_{V}={[V(0)]/[A_{1}(0)]}=1.695±0.083±0.051, r_{2}={[A_{2}(0)]/[A_{1}(0)]}=0.845±0.056±0.039.