I am trying to find the motion's equation for a charged particle $q$ with mass $m$ in an EM field. First the 2 equations I have are: $$\frac{\mathrm dU_x}{\mathrm dt}= W_cU_y + \frac{W_cE_x}{B_z}\\ \frac{\mathrm dU_y}{\mathrm dt}= -W_cU_x + \frac{W_cE_y}{B_z}$$ After I use a new complex variable $U'= U_x + \mathrm iU_y$, I come up with the differential equation $$\frac{\mathrm dU'}{\mathrm dt} + \mathrm iW_cU'= \frac{W_c}{B_z}(E_x+\mathrm iE_y),$$ which gives us $$U'= u_o*e^i*(-W_c-d) - i*E_x/B_z + E_y/B_z.$$so
$u_x(t)= u_o*\cos(-W_c-d) + E_y/B_z$
$u_y(t)= u_o*\sin(-W_c-d) - E_x/B_z$
I have come up with these 2 by integrate the previous
$X(t)= -u_o/W_c*(\sin(-W_c*t-d) + t*E_y/B_z + x_o$
$Y(t)= u_o/W_c*(\cos(-W_c*t-d) - t*E_x/B_z +y_o$
uo,xo,yo,d,Ex,Ey,B are known obv and $W_c=\dfrac{qB_z}{m}$.
please any help would be nice!