Separable Equations: Answer to Example 1

Example: Find all solutions to


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


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


Then we integrate



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

we get


Therefore, we have


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


Next Example.

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

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