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)