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My text book has introduced short explanatory note as following:

"For given differential equation and its implicit solution, if you change it into a form of explicit one, there's a possibility the latter one would not satisfy the initial condition."

However, there's no example given.

Far now, I just had learned 1st-order differential equation which could be solved by separation of variables approach.

Any example falls into the above note?

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Consider for example the DE $$ y' = \frac{t}{y}, \quad y(1) = 2. $$ Solving using separation of variables, we find that an implicit solution is $$ y^2 = t^2 + 3, $$ which indeed satisfies the initial condition. Now, rewriting as an explicit solution, i.e. $$ y(t) = \pm \sqrt{t^2 + 3}, $$ we run into a problem, namely that we have two candidate solutions. However, note that the negative candidate gives $y(1) = -2$, which clearly violates the initial condition (I suspect this is what your text book is describing).

The positive solution satisfies our initial condition, hence the solution we are looking for is $$ y(t) = \sqrt{t^2 + 3}. $$

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  • $\begingroup$ thx. I think your suspection is exact what it describes. $\endgroup$ – Daschin Jul 18 '17 at 8:24
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    $\begingroup$ If that is what the author was hinting at, I would consider it bad writing at best. Actually, given the vagueness of the assertion, I think it's bad writing at any rate. $\endgroup$ – Harald Hanche-Olsen Jul 18 '17 at 8:25
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    $\begingroup$ @HaraldHanche-Olsen I concur. $\endgroup$ – ekkilop Jul 18 '17 at 8:27
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    $\begingroup$ @Daschin Sorry, no easy answers to that one very general question! I can only give the standard answer: Practice math. And perhaps one more: Once you have tackled a problem, don't leap immediately into the next one, but spend some moments reviewing what you did. Where did you get stuck, if you did, and why? Can you think of a different way to arrive at the same answer? How does it fit into the surrounding landscape of similar problems? With time, mathematics will seem more like a unified whole instead of just a random collection of tricks. $\endgroup$ – Harald Hanche-Olsen Jul 18 '17 at 8:43
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    $\begingroup$ (continue) but this kind of approach was helpful for me while I was in highschool but it looks more and more useless and something paradoxcially unpractical to actually utilize mathmatics. If you have experienced or see the student of yours who suffrered from similar problms, please let me advised. $\endgroup$ – Daschin Jul 18 '17 at 8:48

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