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General Solution: A solution of a differential equation in which the number of arbitrary constants is equal to the order of the differential equation is called the general solution or complete integral or complete primitive.

Singular solution: A solution which can not be obtained from a general solution is called singular solution.

I have no objection on this two definition, both satisfy the given ODE and are clear to me and I have the idea how to find the general solution and the singular solution of an ODE.

My question is:

Although singular solution does not obtain from general solution (what is given in the definition), yet why it is called "general"?

I think my question is not a duplicate. If so please forgive me.

Thanks for your valuable time.

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    $\begingroup$ A slightly informal explanation is that the general solution is the most restrictive form possible for "most" of the possible initial value conditions, but sometimes there might be a couple of particular isolated points that you could set for the initial value that would cause the solution to not be unique, at which point the "general solution" might still hold; but so might a lot of other solutions! You can find a few concrete examples of this occurring on Wikipedia (en.wikipedia.org/wiki/Singular_solution#Failure_of_uniqueness) $\endgroup$
    – Jack Crawford
    May 24, 2019 at 10:07
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    $\begingroup$ It's like if I asked you to solve $0 = ax^2 + bx + c$ for $x$; you would probably tell me to use the quadratic formula, because it is the "general solution" -- but if I then turned around and told you that $a=0$, the quadratic formula would break, and you'd see that it being the "general solution" isn't exactly the same thing as meaning that it holds always. $\endgroup$
    – Jack Crawford
    May 24, 2019 at 10:10
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    $\begingroup$ One should probably emphasize that there may be more than one "general solution" of this definition. Like $y=\frac{Ae^x-Be^{-x}}{Ae^x+Be^{-x}}$ give both for $A=1$ and $B=1$ general solutions to $y'=1-y^2$. The constant solutions $y=\pm 1$ are embedded in only one of these general solutions each. However, they are not singular. $\endgroup$
    – Lutz Lehmann
    Jun 14, 2019 at 8:36

1 Answer 1

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If the ODE with its boundary conditions, does not have a unique solution, then there are sometimes families of solutions, so it would be a general solution as opposed to the general solution. These usually get referred to as families anyway.

If the ODE with its boundary conditions, does have a unique solution, then the singular solutions to all possible combinations of boundary conditions can be written in the form of the general solution.

At least, this is how I understand it.

I don't know if that clears it up for you?

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