Summary: This paper investigates the modular differential equation \(y^{\prime\prime} + sE_4y = 0\) on the upper half-plane \(\mathbb{H}\), where \(E_4\) is the weight 4 Eisenstein series and \(s\) is a complex parameter. This is equivalent to studying the Schwarz differential equation \(\{h, \tau\} = 2sE_4\), where the unknown \(h\) is a meromorphic function on \(\mathbb{H}\). On the other hand, such a solution \(h\) must satisfy \(h(\gamma\tau) = \varrho(\gamma)h(\tau)\) for \(\tau \in\mathbb{H}\), \(\gamma \in\mathrm{SL}_2(\mathbb{Z})\) and \(\varrho\) being a 2-dimensional complex representation of \(\mathrm{SL}_2(\mathbb{Z})\). Moreover, in order for \(h\) to be meromorphic or to have logarithmic singularities at the \(\mathrm{SL}_2(\mathbb{Z})\)-cusps of \(\mathbb{H}\), it is necessary to have \(s = \pi^2r^2\) with \(r\) being a rational number. For \(r = m/n\) in reduced form, it turns out that \(\varrho\) is irreducible with finite image if and only if \(2 \leq n \leq 5\) and in this case \(h\) is a modular function for the genus zero torsion-free principal congruence group \(\Gamma(n)\), while \(\varrho\) is reducible if and only if \(n = 6\). By Solving an explicit algebraic system, we prove that solutions for any \(r = m/n\) can be built from a solution corresponding to \(r = 1/n\), for \(2 \leq n \leq 6\), by integrating certain weight 2 meromorphic modular forms. Together with the earlier work by the authors for \(r\) being an integer [20], this provides the solutions to the above-mentioned differential equations for all \(r = m/n\) with \(1 \leq n \leq 6\).