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Questions tagged [singular-solution]

For questions about the singular solution of the ordinary differential equations. It is a special type of solution different from general solution. Such solutions does not contain any arbitrary constant and is not a particular case of the general solution.

Suppose $f(x,y,c)=0$ is the solution of $\phi(p,x,y)=0$ where $p=\frac{dy}{dx}$.$\quad \phi(p,x,y)=0$ defines the slope of the tangent of that curve of the family which passes through any point $(x,y)$, which we choose, which is the value for $p$ for that particular point.

In other words the tangent and the curve of the family have a common value of $x,y,p$. We have just noticed above that the tangent of any point of the envelope of the family of curves coincides with that of the curve through that point.

Therefore, the equation of the envelope will satisfy the differential equation and is a solution which is different from the general solution, for we can not obtain its equation by assigning a certain value to the parameter. This solution is called the singular solution.

Reference:

"https://archive.org/details/singularsolution00ward"

"https://en.wikipedia.org/wiki/Singular_solution"

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