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?
Best Answer
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 $$ y(t) = \sqrt{t^2 + 3}. $$