Electron movement in conductors

[anachin6000]

This is not a normal problem help topic. The difficulty I've encountered is in understanding an alternative solution.

1. Homework Statement
In a real conductor, electrons (with mass m), conducted by external electric fields, constantly collide with defects and impurities within the conductor. The average effect of those collisions is similar to a viscosity force ƒ= -mv/τ (ƒ and v have vector hats), where τ is a constant parameter, named collision time.
1. Write the second law of dynamics that helps finding the speed of an electron. Ignore the interactions between electrons.
2. Consider that E(t) = E₀ sinωt (E and E₀ have vector hats), find the expression for the speed of an electron. Note: it's convenient to use the complex form E(t) = E₀ e^iωt. You can use e^iθ = cosθ +i sinθ.

...(there are other tasks, but they are rather easy)

Homework Equations

ma +mv/τ = qE₀ sinωt (a, v, E do not have vector hats)

ma= -mv/τ - qE₀ e^iωt (a, v, E are vectors)

The Attempt at a Solution

I can solve the first differential equation (that is the equation I have derived) the classical way (find homogeneous and particular solutions). Tough, in the key, they write the equation in the second form and then they get the next equation:

iωmv= -mv/τ - qE₀ (v, E are vectors)

What is the explanation for their equation. I just can't get it.

 

[SophiaSimon]

I can solve the first differential equation (that is the equation I have derived) the classical way (find homogeneous and particular solutions). Tough, in the key, they write the equation in the second form and then they get the next equation:

iωmv= -mv/τ - qE0 (v, E are vectors)

What is the explanation for their equation. I just can't get it.

If you solve your above equation for velocity, then you get a function that doesn't change with time. How can you have an acceleration if your velocity is constant with respect to time?