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unsolved equation

asked 2011-08-08 05:48:57 -0500

milofis gravatar image

updated 2011-08-08 14:25:13 -0500

Dear sage user,

Could you help me to find z = z(t)?, if:

((2.867e28) / (sqrt (r^3 - (2.1e13)^3 )) ) == (( (2.121e8)*(sqrt(2*(z^2) -1))) / ( z - sqrt (z^2 - 1)))


r == ((1.97774e13)*(sqrt(sqrt(t)))) + ((3.55214e12)*(sqrt(t)))

1 < t < 6000

z(t= 1) = 100

r (t= 1) = 2.332954e13


I couldn't solve it!!!

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This isn't a system of linear equations, nor a system of ordinary differential equations, unless you've made a typo. Try rephrasing your question, or looking at the Sage documentation about solving linear or differential equations.

benjaminfjones gravatar imagebenjaminfjones ( 2011-08-08 07:16:59 -0500 )edit

could you tell me,what kind of equation do you think is it?. I found this equation from physical principles of shock wave to describe the temporal evolution of relativistic blastwaves. z is the lorentz factor ( ) and r is the shock radius. Thank you for your comment!

milofis gravatar imagemilofis ( 2011-08-08 14:37:44 -0500 )edit

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answered 2011-08-08 16:09:38 -0500

benjaminfjones gravatar image

A differential equation would involve the derivative z'(t) which you don't have in your equation(s).

If you substitute the value of r into the first equation you would have a single non-linear equation relating z and t. You may be able to solve this equation using the solve command. You don't have any unknown parameters in the expression for r or the equation involving z so you might not be able to solve the equation and have the values at t=1 that you specify be true.

You might try something like:

r,z,t = var('r,z,t')
equation = ((2.867e28) / (sqrt (r^3 - (2.1e13)^3 )) ) == (( (2.121e8)*(sqrt(2*(z^2) -1))) / ( z - sqrt (z^2 - 1)))
equation = equation.subs(r == ((1.97774e13)*(sqrt(sqrt(t)))) + ((3.55214e12)*(sqrt(t))))
solve(equation, t)

When I tried this I got:

Is  (2*z^2-1)*(z-sqrt(z^2-1))  positive, negative, or zero?

meaning that Maxima (the solver underneath Sage) can't solve the equation without more information.

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Thank you very much for your kind attention and your suggestions. However I need to find z as a function of t, when t vary 1 - 1000 with step of 10. is it possible to do that?

milofis gravatar imagemilofis ( 2011-08-15 17:53:23 -0500 )edit

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Asked: 2011-08-08 05:48:57 -0500

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Last updated: Aug 08 '11