Question : Define an ordinary point, singular point and regular singular point of a differential equation.singular point and regular singular point of a differential equation?
Solution:
Lets consider a second order linear homogeneous differential equation \[y^{\prime\prime}\ +a_1\left(x\right)y^\prime+a_0(x)y(x)=0 \] (1)
Ordinary point
Any point x0 is said to be an ordinary point of the eqn. (1) If both the coefficients \[a_1\left(x\right),a_0\left(x\right)\] are analytic at x=x0
Examples
x = 0 is an ordinary point of the differential equation \[y^{\prime\prime}-\frac{1}{\left(1+x^2\right)}y=0 \] x =1 is an ordinary point of the differential equation \[\left(x^2+1\right)y^{\prime\prime}+2xy^\prime+y=0\]
Singular point
Any point x0 is said to be a singular point of the eqn. (1) If one or both the coefficients \[a_1\left(x\right),a_0\left(x\right)\] are not analytic at x=x0.
Examples
x = 0 is a singular point of the differential equation\[ y^{\prime\prime}+\frac{1}{x}y=0\]
x =-1 is a singular point of the differential equation \[\left(x+1\right)y^{\prime\prime}+xy^\prime+y=0\]
Regular singular point
Any point x0 is said to be a regular singular point of the eqn. (1) if x0 is a singular point of the eqn (1) and following limits are finite. \[\lim_{x\rightarrow x0}{(x-x_0)a_1\left(x\right)}\ ,\lim_{x\rightarrow x0}{\left(x-x_0\right)^2a_0(x)}.\] Otherwise a singular point is said to an irregular singular point.
Examples :
x = 0 is a Regular singular point of the differential equation \[{x^2y}^{\prime\prime}+xy^\prime+y=0 \]
x =2 is a Regular singular point of the differential equation \[\left(x-2\right)^2y^{\prime\prime}-2\left(x+2\right)y^\prime-y=0\]