User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

DiffEQ homework help
Hello friends! Sorry I haven't contributed in a while, I haven't had much to contribute, but now I'm learning diff.eq. and hopefully will be able to contribute to that area. In the meantime, I'm having a hard time with this homework problem and would appreciate your help. I'm not sure if I'm making a mistake somewhere, but I don't know how to continue.

Let $\gamma, T$ be constants. Solve for $y$:


 * $\dfrac {\mathrm dy}{\mathrm dt} = (\gamma \cos t + T)y - y^3$

If $y = 0$, $\dfrac {\mathrm dy}{\mathrm dt} = 0$. Else,

Let $v = y^{-2}$. Then $\dfrac {\mathrm dy}{\mathrm dt} = \dfrac {-1} 2 y^3 \dfrac {\mathrm dv}{\mathrm dt}$

So $y^3\dfrac {\mathrm dv}{\mathrm dt} + 2(\gamma \cos t + T)y = -y^3$


 * $\implies y^3\dfrac {\mathrm dv}{\mathrm dt} + 2(\gamma \cos t + t)v = 2$

Let $\mu = \exp(-2\gamma\sin t + T^2)$, then $\dfrac {\mathrm d\mu}{\mathrm dt} = (2\gamma\cos t + 2T)\mu$

Multiply the above eqn by $\mu$, then


 * $\mu \dfrac {\mathrm dv}{\mathrm dt} + \dfrac {\mathrm d\mu}{\mathrm dt}v = 2\mu$


 * $\implies D_t(\mu v)= 2\mu$

And from there I don't know how to integrate the RHS WRT t? Did I do something wrong?

--GFauxPas (talk) 21:55, 4 March 2014 (UTC)


 * There are some small flaws (sign error at the $\implies$ sign, $T \ne t$ is a constant, $\cos$ is derivative of $\sin$), but Mathematica is unable to solve this as well. Any known constraints for $\gamma, T$? Perhaps you're asked to do a power series solution? &mdash; Lord_Farin (talk) 00:21, 5 March 2014 (UTC)


 * Excuse my mistakes, I did it in a rush. But here is the question: http://imageshack.com/a/img401/1748/5a6j.jpg --GFauxPas (talk) 00:40, 5 March 2014 (UTC)


 * It's still an inhomogeneous linear equation, and indeed it seems that the integral you are left with is the relevant one. Perhaps it's permissible to leave an integral sign in your answer? Because it cannot be done in elementary terms (dixit Mathematica). &mdash; Lord_Farin (talk) 08:51, 5 March 2014 (UTC)


 * Perhaps the purpose of the exercise was just to show that the substitution turns the equation into a linear one. --GFauxPas (talk) 12:19, 5 March 2014 (UTC)