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I want to know that, is there any necessary and sufficient condition for a system of differential equation

$$ \frac{dx_i}{dt} = c_i f$$ where $f$ is a smooth function from $\mathbb R^n$ to $\mathbb R$ and $c_i$ are constant for $i=1,2...n$; to have a unique solution.

Thanks in advance.

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I assume that $c_i$ are constant? – Tunococ Sep 7 '12 at 3:51

Let $y_i=x_i/c_i$ and $g(y)=f(c_1y_1,\dots,c_ny_n)$. Then the system can be written as $$ \frac{dy_i}{dt}=g(y),\quad1\le i\le n.\tag{1} $$ If $\{y_1(t),\dots f_n(t)\}$ is a solution, then there exist constants $k_2,\dots,k_n$ such that $y_i=y_1+k_i$, $2\le n$. Conversely, if the real valued function $z(t)$ satisfies de differential equation $$ \frac{dz}{dt}=g(z,z+k_2,\dots,z+k_n)\tag{2} $$ for some constants $k_2,\dots,k_n$, then $y=\{z,z+k_2,\dots,z+k_n\}$ is a solution of (1).

For any choice of constants $k_2,\dots,k_n$ and $\zeta\in\mathbb{R}$, since $g$ is smooth, equation (2) has a unique solution such that $z(0)=\zeta$. This gives you an infinite number of different solutions of (1).

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