Q - 9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Question: In Problems 1–10, determine all the singular points of the given differential equation.

9. (sin)y"-(in)y=0

Step-by-Step Solution

Answer

The singular point exists in this differential equation for ^Q(x)  is at ⁼ⁿ for ⁰ where ^n=1,2,3.....

Step 1: Ordinary and Singular Points

A point  is^x₀ called an ordinary point of equationy'+p(x) y'+q(x)y = 0 if both p and q are analytic at ^X 0 . If^x0 is not an ordinary point, it is called a singular point of the equation

Step 2: Find the singular points

The given differential equation is

 (sin)y"-(in)y=0

Dividing the above equation by (sin) 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 ^Q(X) this occurs at  which implies

We see that ^Q(X)  is actually analytic at  which implies

Therefore, ^Q(x) is actually except at  for  

The singular point exists in this differential equation for ^Q(x)  is at ⁼ⁿ for ⁰ where n=1,2,3.....