Update 3_drude_model_solutions.md - typo fix

src/3_drude_model_solutions.md

@@ -67,22 +67,22 @@ This equation describes a circular motion around the point $x_0=x(0), y_0=y(0)+v
 
 Due to the applied electric field $\bf E$ in the $x$-direction, the equations of motion acquire an extra term:
 $$
-m v_x' = -e(E_x + v_yB_z)
+m v_x' = -e(E_x + v_yB_z).
 $$
-This leads to the same 2nd-order D.E. for v_x as above. However, for $v_y$ we get
+Differentiating w.r.t. time leads to the same 2nd-order D.E. for $v_x$ as above. However, for $v_y$ we get
 $$
 v_y'' = -\omega_c^2(\frac{E_x}{B_z}+v_y),
 $$ 
 which has as solution 
 $$
-v_y(t) = v_0\sin(\omega_t)-\frac{E_x}{B_z}.
+v_y(t) = v_0\sin(\omega_c t)-\frac{E_x}{B_z}.
 $$
 By integrating the expressions for the velocity we find:
 $$
-(x(t)-x_0)^2 + (y(t) - y_0 + \frac{E}{B}t))^2 = \frac{v_0^2}{\omega_c^2}
+(x(t)-x_0)^2 + (y(t) - y_0 + \frac{E_x}{B_z}t))^2 = \frac{v_0^2}{\omega_c^2}.
 $$
 
-This represents a [cycloid](https://en.wikipedia.org/wiki/Cycloid#/media/File:Cycloid_f.gif): a circular motion around a point that moves in the $y$-direction with a constant velocity $\frac{E}{B}$.
+This represents a [cycloid](https://en.wikipedia.org/wiki/Cycloid#/media/File:Cycloid_f.gif): a circular motion around a point that moves in the $y$-direction with velocity $\frac{E_x}{B_z}$.
 
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