Laplace Analysis of Step Response of a Parallel RC Circuit
My solution for m = 1 case:

Few points:

As @Chu said above, units inside the KCL/KVL equations can really confuse, I offer you to just work with your preferred unit system (SI / cgs).

Pay attention to the currents direction when you do KCL. In KCL, the sum of the currents going in the node should be equal to the sum of the currents going out from the node. You equated two currents going out from the node, while the should be in the same side of the equation (C*dVo/dt = Vo/10k).