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We apply the BPS Lagrangian method~\cite{Atmaja:2015umo} to derive BPSequations of monopole and dyon in the $SU(2)$ Yang-Mills-Higgs model,Nakamula-Shiraishi models, and their Generalized versions. We argue that byidentifying the effective fields of scalar field, $f$, and of time-componentgauge field, $j$, explicitly by $j=\beta f$ with $\beta$ is a real constant,the usual BPS equations for dyon can be obtained naturally. We validate thisidentification by showing that both Euler-Lagrange equations for $f$ and $j$are identical in the BPS limit. The value of $\beta$ is bounded to $|\beta|<1$due to reality condition on the resulting BPS equations. In the Born-Infeldtype of actions, namely Nakamula-Shiraishi models and their Generalizedversions, we find a new feature that adding the energy density by a constant$4b^2$, with $b$ is the Born-Infeld parameter, will turn monopole(dyon) toanti-monopole(anti-dyon) and vice versa. In all Generalized versions there areadditional constraint equations that relate the scalar-dependent couplings ofscalar and of gauge kinectic terms; or $G$ and $w$ respectively. For monopolethe constraint equation is $G=w^{-1}$, while for dyon is $w(G-\beta^2w)=1-\beta^2$ which further gives lower bound to $G$ as such$G\geq|2\beta\sqrt{1-\beta^2}|$. We also write down the complete square-formsof all effective Lagrangians.

BPS Equations of Monopole and Dyon in $SU(2)$ Yang-Mills-Higgs Model,  Nakamula-Shiraishi Models, and Their Generalized Versions from The BPS  Lagrangian Method