Analysis of a circuit with transformer, R and C components
You appear to be doing a transient analysis; this problem calls for complex notation. Represent the capacitor's reactance as: \$\small -j\:X_C=-\large\frac{j}{\omega C}\small =-j\$, let the source voltage be: \$\small V_S=4+j0=4\$, and then use nodal analysis.