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By analyzing $e^+e^-$ annihilation data corresponding to an integratedluminosity of $2.93\,\rm fb^{-1}$ collected at the center-of-mass energy of3.773\,GeV with the BESIII detector, we report the first observations of thedoubly Cabibbo-suppressed decays $D^+\to K^+\pi^0\pi^0$ and $D^+\toK^+\pi^0\eta$. The branching fractions of $D^+\to K^+\pi^0\pi^0$ and $D^+\toK^+\pi^0\eta$ are measured to be $(2.1 \pm 0.4_{\rm stat} \pm 0.1_{\rmsyst})\times 10^{-4}$ and $(2.1 \pm 0.5_{\rm stat} \pm 0.1_{\rm syst})\times10^{-4}$ with statistical significances of 8.8$\sigma$ and 5.5$\sigma$,respectively. In addition, we search for the subprocesses $D^+\toK^{*}(892)^{+}\pi^0$ and $D^+\to K^{*}(892)^{+}\eta$ with $K^{*}(892)^+\toK^+\pi^0$. The branching fraction of $D^+\to K^{*}(892)^{+}\eta$ is determinedto be $({4.4^{+1.8}_{-1.5}}_{\rm stat}\pm0.2_{\rm syst})\times10^{-4}$, with astatistical significance of 3.2$\sigma$. No significant signal for $D^+\toK^{*}(892)^{+}\pi^0$ is found and we set an upper limit on the branchingfraction of this decay at the 90\% confidence level to be $5.4\times10^{-4}$.

Observation of the doubly Cabibbo-Suppressed decays $D^+\to K^+\pi^0\pi^0$ and $D^+\to K^+\pi^0\eta$