In the present work the initial value problem of the equation \[ D^ k_ t u=\sum^{k}_{j=1}a_ jD_ t^{k-j}(-1)^{m+1} \nabla^{2m} u+\sum^{k-1}_{j=0}\Lambda_ j(t)D^ j_ t u \] where \((A_ j(t)\), \(j=0,1,...,k-1\), \(0\leq t\leq T)\) is a family of bounded linear operators defined on \(C(R_ n)\), the space of all continuous functions defined on \(R_ n\) with the norm \(\| f\| =\max_{x\in R_ n} | f(x)|\), \(f\in C(R_ n)\) is considered. It is found a suitable formula for solution, the uniqueness of the solution is also proved and the correct formulation of the Cauchy problem for this equation will be studied.