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I have a set of ODE where

1) [tex]\frac{dv_x}{dt}=\frac{q(t)B}{m}v_y[/tex]

2) [tex]\frac{dv_y}{dt}=\frac{q(t)B}{m}v_x[/tex]

3) [tex]\frac{dv_z}{dt}=0[/tex]

Following the strategy to solve a simple harmonic oscillator,

I differentiate (1) to get

4) [tex]\frac{d^2v_x}{dt^2}=\frac{q(t)B}{m}\frac{dv_y}{dt}+q'(t)v_y[/tex]

and substitute (1) and (2) into it to get

5) [tex]\frac{d^2v_x}{dt^2}=(\frac{q(t)B}{m})^2v_x+\frac{q'(t)}{q(t)}\frac{dv_x}{dt}[/tex]

which involve only functions of [itex]t[/itex] and [itex]x[/itex].

The initial position of the particle is at the origin.

I do not know if this is the correct way. I should solve it by inspired guessing?

Can you please help me by giving me some tips on solving this?

Thank you very much.

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# Solving Nonlinear ODE: magnetism with varying particle charge

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