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Differential Equations

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Moderate

First-order Scalar Autonomous Differential Equation

DIFFEQ-NJXETL

A first-order scalar autonomous differential equation is an equation of the form $x'=f(x)$, where the unknown is $x=x(t)$. Which of the following properties is (are) shared by all autonomous differential equations?

Assume that we only consider equations $x'=f(x)$ for which $f$ is continuously differentiable.

Select ALL correct answers.

A

If $x$ is a solution, then $x$ is independent of $t$.

B

If $x$ is a solution and $c$ is a constant, then $y$ defined by $y(t)=x(t+c)$ is also a solution.

C

If $x$, $y$ are two different solutions of $x'=f(x)$ and $x(t_1)=y(t_2)$ for some $t_1\neq t_2$, then $y(t)=x(t+c)$ for some constant $c$.

D

If $x$ is the constant function $x(t)=c$ and $f(c)=0$, then $x$ is a solution of $x'=f(x)$.