Simple derivations in two variables

$ $Let$k$beafieldofcharacteristiczero\mathrm{.}If$c_1, c_2\in k\setminus {0}, s,t\geq 1$and$u\geq 0$,thenitisshownthatthe$k$-derivations$\partial_x + x^u(c_1x^ty^s+c_2)\partial_y$and$\partial_x + x^u(c_1x^t+c_2y^{s+1})\partial_y$of$k[x,y]$aresimple\mathrm{.}Wealsogiveanecessaryandsufficientconditionforthe$k$-derivation$y^r\partial_x + (c_1x^{t_1}y^{s_1}+c_2x^{t_2}y^{s_2})\partial_y$,where$r, t_1, s_1, t_2, s_2 \geq 0$and$c_1, c_2\in k$,of$k[x,y]$ to be simple.