Find Singular Solution to dy/dx=x(1-y^2)^(1/2)

[andrewdavid]

Here is the differential equation I have: dy/dx=x(1-y^2)^(1/2) . I'm supposed to find a singular solution to that equation but I'm really not sure how to figure it out. I can separate and integrate it, but then what is the singular solution?

 

[saltydog]

Here is the differential equation I have: dy/dx=x(1-y^2)^(1/2) . I'm supposed to find a singular solution to that equation but I'm really not sure how to figure it out. I can separate and integrate it, but then what is the singular solution?

For the equation:
y^{'}=x\sqrt{1-y^2}

As you stated, you can separate variables, integrate, and find y(x). However, some non-linear equations have "other" solutions which are not derived from this general solution. These are called "singular solutions". Note you can divide by the radical assuming it's not zero. If it is zero, then:

1-y^2=0

But if that's so, then what does that tell you y(x) has to be? Well, + or - 1 right? Are those derived from the solution you get when you separate variables and integrate? Singular solutions "envelop" general solutions. Check that out. Plot some examples of the general solutions and the singular solutions and see what I mean.

 

[saltydog]

So how about a plot showing this Andrew? You know a plot really makes this all clear especially to others that may be new to this. I guess if you don't then I'll submit one in a day or so.

 

[saltydog]

Attached is a plot of some examples of particular solutions of:

y(x)=\sin(\frac{x^2}{2}+c)

and the two singular solutions y(x)=1 and y(x)=-1. Note how the singular solutions envelop the particular solutions.