Explanation of the Node-Voltage Method equation in specific circuit
The only difference between the given equation for \$u_C(t)\$ and your equation is the right hand side is multiplied by -1. Since the dot of the primary side of transformer is at the top and the dot for the secondary side is at the bottom, the polarity of the secondary side will be opposite that of the primary side. 

http://en.wikipedia.org/wiki/Polarity_(mutual_inductance) 

Had both the dots been at the top, your equation would be right.