Separable Equations: Answer to Example 1

Example: Find all solutions to

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Solution: First, we look for the constant solutions, that is, we look for the roots of

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This equation does not have real roots. Therefore, we do not have constant solutions.

The next step will be to look for the non-constant solutions. We proceed by separating the two variables to get

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Then we integrate

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Since

$\displaystyle {\frac{{1}}{{1 + \displaystyle \frac{1}{y^2}}}}$ = $\displaystyle {\frac{{y^2}}{{y^2+1}}}$ = 1 - $\displaystyle {\frac{{1}}{{y^2+1}}}$

we get

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Therefore, we have

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It is not easy to obtain y as a function of t, meaning finding y in an explicit form.

Finally, because there are no constant solutions, all the solutions are given by the implicit equation

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Next Example.

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Author: Mohamed Amine Khamsi
Last Update 6-22-98

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