  In this paper, we define the normalized Eisenstein series $\mathcal{P}$, $e$,
and $\mathcal{Q}$ associated with $\Gamma_0(2),$ and derive three differential
equations satisfied by them from some trigonometric identities. By using these
three formulas, we define a differential equation depending on the weights of
modular forms on $\Gamma_0(2)$ and then construct its modular solutions by
using orthogonal polynomials and Gaussian hypergeometric series. We also
construct a certain class of infinite series connected with the triangular
numbers. Finally, we derive a combinatorial identity from a formula involving
the triangular numbers.
