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Differential Equations

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Properties of Solutions for Separable Nonlinear Equations

DIFFEQ-YS1E5Y

Consider the equation $y'+y^2=1$. Which of the following statements are WRONG?

Select ALL that apply.

A

The full list of solutions is $y=(Ce^{2x}+1)/(Ce^{2x}-1), C\in\mathbb{R}$ or $y=1$.

B

The full list of solutions is $y=(Ce^{2x}-1)/(Ce^{2x}+1), C\in\mathbb{R}$ or $y=1$.

C

By the formula for the solutions, all solutions except $y=-1$ converge to $1$ as $x\to+\infty$.

D

Since $y=-1$ is a solution, the solution curves under $y=-1$ never cross $y=-1$ by the Existence and Uniqueness Theorem.

E

The solution satisfying $y(0)=-2$ is $y(x)=(e^{2x}+3)/(e^{2x}-3)$. As $x\to+\infty$, the limit is $1$ and hence the curve crosses $y=-1$.