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December 30th, 2018, 02:45 AM  #1 
Newbie Joined: Dec 2018 From: netherlands Posts: 1 Thanks: 0  Thought problem differential equation
Does anybody know how to solve the following differntial equation? $\displaystyle ω^2⋅r+dω/dt⋅r+dr/dt⋅ω=g$ where w and r are a function of time so ω(t) and r(t). g is a constant This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem. Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant. The vehicle will have 2 acceleration components, a radial component ($\displaystyle a_r$) and a change in speed component ($\displaystyle a_s$). Since we want constant acceleration: ar+as=constant From circular motion we know that $\displaystyle a_r=ω^2r$ where r is the radius, and ω Further we know that v=ω∗r, therefore: $\displaystyle a_s=d/dt(ωr)$ For constant acceleration $\displaystyle a_{total}=a_r+a_s=ω^2r+d/dt(ωr)=constant$ Further usefull information: Length of the track covered at a certain time l=v∗t=ωrt With all this the radius will be a function of time, r(t) and the angular velocity will be a function of time, ω(t) Boundary conditions: r(0)=0 ω(0)=0 Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $\displaystyle v(t_{end})=100[m/s]$ $\displaystyle and l(t_{end})=5000[m] $ 
December 30th, 2018, 10:28 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,429 Thanks: 1314 
What I would do is parameterize your curve. From that parameterization the acceleration components are easily derived and then you will have a much less general differential equation. As you've written it $\omega(t),~r(t)$ are much too general to do anything with. 

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