Question: In Problems 1–10, determine all the singular points of the given differential equation.
4. (x2+x)y"+3y'-6xy = 0
The singular point exists in this differential equation for both P(x) and Q(x) is at x = 0,-1
A point is called an ordinary point of equation y"+p(x)y'+q(x)y = 0 if both p and q are analytic at . If is not an ordinary point, it is called a singular point of the equation.
The given differential equation is,
(x2+x)y"+3y-6xy = 0
Dividing the above equation by (x2+x)we get,
On comparing the above equation with y"+p(x)y'+q(x)y = 0, we find that,
Hence, P(x) and Q(x) are analytic except, perhaps, when their denominators are zero.
For P(x) this occurs at x = 0,-1 .
We see that Q(x) is actually analytic at x = 0 .
Since we can cancel an x in the numerator and denominator of Q(x) as shown in (1).
Therefore, P(x) is analytic except at x = 0,-1 as well as Q(x) is analytic except at x = -1 .
The singular point exists in this differential equation for both P(x) and Q(x) is at x = 0,-1 .
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