The　homotopy　continuation　method　has　been　widely　used　to　compute　multiple　solutions　of　nonlinear　differential　equations,　but　the　computational　cost　grows　exponentially　based　on　the　traditional　finite　difference　and　finite　element　discretizations.　In　this　work,　we　presented　a　new　method　by　constructing　a　spectral　approximation　space　adaptively　based　on　a　greedy　algorithm　for　nonlinear　differential　equations.　Then　multiple　solutions　were　computed　by　the　homotopy　continuation　method　on　this　low-dimensional　approximation　space.　Various　numerical　examples　were　given　to　illustrate　the　feasibility　and　the　efficiency　of　this　new　approach.