1. The problem statement, all variables and given/known data
Solve the equations of motion ##\ddot{y}= \omega \dot{z}## and ##\ddot{z}= \omega (\frac{E}{B}-\dot{y})##
2. Relevant equations
3. The attempt at a solution
Integrate the first equation to get ##\dot{y}=\omega z + c_1## and plug into equation 2: ##\ddot{z}=\omega (\frac{E}{B}-\omega z + c_1## simplify ##\ddot{z}= \omega \frac{E}{B} - \omega ^2 z + \omega c_1## and integrating again leaves ##\dot{z}= \omega \frac{E}{B}t - \omega ^2 zt + c_1 t +c_2##
The next step is where everything seems to go off the rails. second integration of ##\dot{y}## gives ##\omega z t + c_1 t + c_3 = y(t)## and the second integral of ##\dot{z}## gives ##\omega \frac{E}{B}t-\omega ^2 z t + \omega c_1 t + \omega c_4##
The solution the worked example gives are $$ y(t) = c_1 cos(\omega t) + c_2 sin(\omega t) + (E/Bt + c_3$$ and $$z(t) = c_2 cos(\omega t) - c_1 sin(\omega t) + c_4$$
Where did the trig functions come from?!
Solve the equations of motion ##\ddot{y}= \omega \dot{z}## and ##\ddot{z}= \omega (\frac{E}{B}-\dot{y})##
2. Relevant equations
3. The attempt at a solution
Integrate the first equation to get ##\dot{y}=\omega z + c_1## and plug into equation 2: ##\ddot{z}=\omega (\frac{E}{B}-\omega z + c_1## simplify ##\ddot{z}= \omega \frac{E}{B} - \omega ^2 z + \omega c_1## and integrating again leaves ##\dot{z}= \omega \frac{E}{B}t - \omega ^2 zt + c_1 t +c_2##
The next step is where everything seems to go off the rails. second integration of ##\dot{y}## gives ##\omega z t + c_1 t + c_3 = y(t)## and the second integral of ##\dot{z}## gives ##\omega \frac{E}{B}t-\omega ^2 z t + \omega c_1 t + \omega c_4##
The solution the worked example gives are $$ y(t) = c_1 cos(\omega t) + c_2 sin(\omega t) + (E/Bt + c_3$$ and $$z(t) = c_2 cos(\omega t) - c_1 sin(\omega t) + c_4$$
Where did the trig functions come from?!
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