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I have a vector $\vec{w}(t)$, whose components are all positive. I have another vector $\vec{v}(t)$ which is the component-by-component square of $\vec{w}$: $v_i(t) = w_i(t)^2$.

Finally, I have an ODE for $\vec{v}$:

$d\vec{v}(t) / dt = M_1 \vec{v}(t) + M_2 \vec{w}(t)$.

$M_1$ and $M_2$ are relatively simple matrices, I can exponentiate them easily. How do I solve this ODE?

These matrices are constant, upper-triangular, and have non-negative entries.

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FYI, this question asks about the scalar equation of the same type. –  Sasha May 17 '12 at 19:03
Do the matrices depend on $t$? –  user31373 May 17 '12 at 19:45
Even for the $2$-dimensional case, I doubt that you'll get closed-form solutions. –  Robert Israel May 17 '12 at 20:14
The 2D case is, I believe, solvable: $w_2(t) = Ce^{(M_1)_{22}/2 t}$ and $dw_1 /dt = (M_1)_{11} w_1 /2 + (M_2)_{11}/2 + (C(M_1)_{12} e^{(M_1)_{22} t} + C(M_2)_{12} e^{(M_1)_{22}/2 t}) / 2w_1$. I believe this is solvable. –  Craig May 17 '12 at 21:02

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