Particle motion ode (1st order nonlinear nonhomog)

[kyze]

hi all,

I've been trying to work this problem out,

$\frac{dv}{dt}-A(B-v)^{1.6}=G$

A, B and G are constants

and Matlab can't give me a solution either. I'm wondering if there is even a solution?

 

[kyze]

Hi,

Just realized I can make an assumption for B = 0. Does this make it solvable?

 

[JJacquelin]

Hi !
The ODe is of the "separables variables" kind (see in attachment) :

 

[kyze]

wow! thanks

 

[JJacquelin]

Sorry, there was a typo at the end of the attached page :

 

[kyze]

Since v represents velocity, can I then integrate the v-equation? Are inv. Beta functions integratable?

Or would it be simpler to determine the x-position by setting up the original equation as a second order derivative as:

d2xdt2−A(B−v)^1.6=G

and as B = 0 then

$\frac{d^2x}{dt^2}+A(\frac{dx}{dt})^{1.6}=G$
with initial conditions
dx/dt(0) = 0

i.e. particle initially at rest

Sorry to ask again, but could you show me this solution? I really struggle with maths.

 

[JJacquelin]

Hi !

If B=0 your basic equation dv/dt−A(−v)^1.6=G implies v<0 or v=0. If not, (-v)^1.6 would not be real and the solution v(t) would not be real, which is not correct on a physical point of view. So v<0 or v=0.
In the general solution given in my preceeding post, let c=0 and B=0 in the formula. Then Y(0)=0 ; Inverse Beta (0) = 0 which leads to v=0.
The expected solution with condition v=0 at t=0 is obtained with c=0 in the formula.

The function v(t) is not a simple Inverse Beta function, but a combination of several functions with the Inverse Beta among them. It's far too complicated for formal integration. I think that the only practical way is numerical integration. If it is that what you want, the simplest way is probably to use a numerical process for solving directly the differential equation.