Determine an actual explicit solution to y′ = t/y; y(2) = -1.
Solution: We already identify by the previous illustration that an implicit solution to this IVP is y2 = t2 - 3. To determine the explicit solution all we require to do is resolve for y(t).
y(t) = +√(t2 - 3)
Here, we've found a problem. Now there are two functions and we merely want one and actually only one will be correct! We can find out the correct function through reapplying the initial condition. Only one of them will suit the initial condition.
In such case we can notice that the "-"solution will be the accurate one. The actual explicit solution is after that y(t) = - √(t2 - 3)
In such case we were capable to determine an explicit solution to the differential equation. It must be noted though that this will not for all time be possible to determine an explicit solution.
Also, notice that in such case we were only capable to find the explicit actual solution as we had the initial condition to assist us find out which of the two functions would be the accurate solution.
Now we've gotten most of the fundamental definitions out of the manner and thus we can move on other topics.