Simple derivations in two variables

Let k be a field of characteristic zero. If c1,c2∈k∖{0},s,t≥1 and u≥0, then it is shown that the k-derivations ∂x+xu(c1xtys+c2)∂y and ∂x+xu(c1xt+c2ys+1)∂y of k[x, y] are simple. We also give a necessary and sufficient condition for the k-derivation yr∂x+(c1xt1ys1+c2xt2ys2)∂y, where r,t1,s1,t2,s2≥0 and c1,c2∈k, of k[x, y] to be simple.