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Differential Equations

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Autonomous Differential Equation

DIFFEQ-6EGW4X

Let ${\bf f}:{\mathbb R}^2\rightarrow {\mathbb R}^2$ be a continuously differentiable function, where ${\mathbb R}^2$ denotes the set of all points ${\bf x}=(x_1,x_2)$ in the Euclidean plane.

Consider the autonomous differential equation ${\bf x}'={\bf f}({\bf x})$.

Choose the BEST answer.

A

For each ${\bf x}_0\in{\mathbb R}^2$, there exists a unique solution ${\bf x}={\bf x}(t)$ with the property that ${\bf x}(t)={\bf x}_0$ for some value of $t$.

B

For each ${\bf x}_0\in {\mathbb R}^2$, there may exist many solutions ${\bf x}={\bf x}(t)$ with the property that ${\bf x}(t)={\bf x}_0$ for some value of $t$. However, all of these solutions have the same trajectory.

C

For each ${\bf x}_0\in {\mathbb R}^2$, if ${\bf f}({\bf x}_0)\neq {\bf 0}$, then there exist many distinct trajectories passing through ${\bf x}_0$.

D

If ${\bf f}({\bf x}_0)={\bf 0}$, then no solution has a trajectory that passes through ${\bf x}_0$.