Complex second order differential equations with vector variables

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I have 3 second order differential equations that need to be solved. I obtained them by separating the velocity vector into x, y and z components so that i can solve them separately. The parent equation and 3 equations can be seen in the image

  • x is displacement
  • v is velocity vector
  • omega is spin
  • w is wind vector
  • derivative is taken w.r.t time

I have the following questions

Since each equation has all x, y and z components of velocity how can they be solved? They are not constant through time so should this be converted into a partial differential equation

What about |v-w| its scalar but changes with time?

How can this equation be solved for position x?