$1.$ An electron with a mass of m and a charge of $-$e is under the influence of a uniform electric and magnetic fields. The magnetic field intensity vector is defined by the expression H $=$ H$0*$nz and the potential of the uniform electric field is given by the expression q $=$ Uo$/2$d$^2*($x$^2+$ y$^2).$ In the expressions Ho$,$ Uo and d are various constants. Depending on the initial conditions stated below, find the electron's location vector components as a function of time, making the necessary explanations and all intermediate operations. Initial conditions: x$(0)=$ a cos$$ y$(0)=$ a sin$,$ z$(0)=0$ and v$(0)=0,$ where a and $$ are constants.