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What is difference between implicit and explicit solution of an initial value problem? Please explain with example both solutions(implicit and explicit)of same initial value problem? Or without example but in some way that is understandable.

thanks

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    Sometimes you solve an differential equation, and the answer is something of the type : $y+x=\sin(xy)$. While you still don't know exactly (i.e. explicitly) what $y$ is, this relation usually yields enough information to answer to many questions about $y$. If you can find the solution as $y=f(x)$, that is always best, but if you cannot, an equation between $x,y$ is still much better than nothing.2011-10-29

3 Answers 3

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As requested:

Let's use the example initial-value problem

$y^\prime y=-x,\qquad y(0)=r, \qquad r\text{ constant}$

One can derive both an implicit and explicit solution for this DE. The implicit solution to this DE is

$x^2+y(x)^2=r^2$

This solution implicitly defines $y(x)$; all we have here is an equation involving $y(x)$. On the other hand, the explicit solution looks like

$y(x)=\pm\sqrt{r^2-x^2}$

and in this case, $y(x)$ is explicitly defined: $y(x)$ is expressed here as an explicit function with $x$ as the only independent variable.


We aren't always this lucky when we solve differential equations that show up in practice. It often happens that we can only be content with an implicit solution (or a parametric solution, which is a somewhat better state of affairs than having just an implicit solution). One famous example is the differential equation that pops up in the brachistochrone problem:

$(1+(y^\prime)^2)y=r^2$

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    @asd, because in some ways normal equations are easier to analyze than differential equations.2017-10-22
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Explicit solution is a solution where the dependent variable can be separated. For example, $x+2y=0$ is explicit because if y is dependent, I can rewrite it as $y=-\frac{x}{2}$ and my y has been separated.

Implicit is when the dependent variable cannot be separated like $\sin(x+e^y)=3y$.

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    Strictly speaking (even as mentioned in the accepted answer) $y+2y=0$ is an implicit solution, which can be converted into the explicit form $y=-\frac{x}{2}$2017-10-31