Singular solution Totally Explained

Everything about Singular Solution totally explained

A singular solution y_s(x) of an ordinary differential equation is a solution that's tangent to every solution from the family of general solutions. By tangent we mean that there's a point x where y_s(x) = y_c(x) and y'_s(x) = y'_c(x) where y_c is any general solution. Usually, singular solutions appear in differential equations when there's a need to divide in a term that might be equal to zero. Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution.

Example

Consider the following Clairaut's equation:
ť

y(x) = x cdot y' + (y')^2 ,!

where primes denote derivatives with respect to x. We write y' = p and then ť

y(x) = x cdot p + (p)^2 ,!

Now, we'll take the differential according to x:
ť

p  = y' = p + x p' + 2 p p' ,!

which by simple algebra yields ť

0 = (2 p + x )p' ,!

This condition is solved if 2p+x=0 or if p'=0.
If p' = 0 it means that y' = p = c = constant, and the general solution is:
ť

y_c(x) = c cdot x + c^2 ,!

where c is determined by the initial value.
If x + 2p = 0 than we get that p = −(1/2)x and substituting in the ODE gives ť

y_s(x) = -(1/2)x^2 + (-(1/2)x)^2 = -(1/4) cdot x^2 ,!

Now we'll check whether this is a singular solution.
First condition of tangency: y_s(x) = y_c(x). We solve ť

c cdot x + c^2 = y_c(x) = y_s(x) = -(1/4) cdot x^2 ,!

to find the intersection point, which is (

-2c, -c^2

).
Second condition tangency: y'_s(x) = y'_c(x).
We calculate the derivatives:
ť

y_c'(-2 cdot c) = c ,!

y_s'(-2 cdot c) = -(1/2) cdot x |_ = c ,!

We see that both requirements are satisfied and therefore y_s is tangent to general solution y_c. Hence, ť

y_s(x) = -(1/4) cdot x^2 ,!

is a singular solution for the family of general solutions ť

y_c(x) = c cdot x + c^2 ,!

of this Clairaut equation:
ť

y(x) = x cdot y' + (y')^2 ,!

Note: The method shown here can be used as general algorithm to solve any Clairaut's equation, for example first order ODE of the form ť

y(x) = x cdot y' + f(y'). ,!

See also caustic (mathematics).

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