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In developing a particle system simulator, I ended up with this apparently innocuous vectorial differential equation:

$m \vec v\,' = -\mu \vec v + \alpha \vec v / |\vec v|$

Where $\vec v = \vec v(t)$, and $m$, $\mu$ and $\alpha$ are constants. Without that pesky $|\vec v|$ at the denominator I wouldn't have any trouble solving it, but like this I honestly have no clue.

At the end of the line, I have the impression that the equation is then transformed to this system of differential equations:

$\begin{cases} m v_x' = -\mu v_x + \alpha v_x / \sqrt{v_x^2 + v_y^2 + v_z^2} \\ m v_y' = -\mu v_y + \alpha v_y / \sqrt{v_x^2 + v_y^2 + v_z^2} \\ m v_z' = -\mu v_z + \alpha v_z / \sqrt{v_x^2 + v_y^2 + v_z^2} \\ \end{cases}$

Not that it gets me any nearer to find a solution.

Also generic suggestions as for which techniques to use in order to try to solve this equation / system would be happily accepted.

PS: for those who are interested, this equation represents the motion of a dot having a costant propulsion pointing in the same direction of its velocity ($\alpha \vec v / |\vec v|$), being slowed down by the friction of the air ($-\mu \vec v$), while the remaining term is obviously $\vec F = m \vec a = m \vec v\,'$.

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As $v'(t)$ and $v(t)$ are linear dependent for each $t$, shouldn't we have $v(t) = \lambda(t)v(0)$ for some function $\lambda\colon[0,T) \to \mathbb R$. The equation for $\lambda$ reads not much better, but its scalar :). – martini May 22 '12 at 15:21

to finish the comment of matrini: if $\vec{v}(t)=f(t)\vec{n}$ for some fixed unit vector $\vec{n}$, and $f>0$, then you get $f'=-\mu f+\alpha$, which gives $f(t)=(f(0)+\alpha t)e^{-\mu t}$. Any solution is of this form (e.g. since we get a solution for every (non-zero) value of $\vec{v}(0)$).

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