First, we establish a general solution of the non-linear Kronecker product system $(P\otimes Q)(t)y'(t)+(R\otimes S)(t)y(t)=f(t,y(t))$ with the help of variation of parameters formula. Finally, we prove existence and uniqueness results for the non-linear Kronecker product system satisfying general boundary condition $Uy=\alpha,by$ using Schauder-Tychonov's and Brouwer's fixed point theorems.