The paper deals with the Cauchy problem for the equation \[ D_xz(x,y)=\sum^N_{j=1}\{a_{j0}(x)z(a_j(x),y)+a_j(x)\cdot D_yz(a_j(x),y)\}+f(x,y), (x,y)\in[0,T]\times\mathbb{R}^n,\tag{1} \] with the initial condition \[ z(0,y)=0,\quad y\in\mathbb{R}^n,\tag{2} \] where \(T>0\), \(a_j\), \(a_{jk}\in C([0,T],\mathbb{R})\), \(j=1,\dots,N\); \(k=0,\dots,n\), \(f\in C(E,\mathbb{R})\), \(a_j=(a_{j1},\dots,a_{jn})\), \(0\leq a_j(x)\leq x\) for \(x\in[0,\tau]\), \(j=1,\dots,N\). The author obtains explicit solutions of the problem (1), (2) in the form of an infinite series.