Given the differential equations:
(sin x)y′′+ xy′+(x − 1/9)
y = 0:
• Determine all the regular singular points for the Eq.;
• Derive the indicial equation corresponding to each regular point;
• Determine the form of two linearly independent solutions near each
of regular singular points.
• Give the behavior of solutions as x → 0.
To solve the given differential equation and answer the questions, we need to determine the regular singular points, derive the indicial equation for each regular point, find the form of two linearly independent solutions near each regular singular point, and determine the behavior of solutions as x approaches 0.
1. Determining the regular singular points:
A singular point is regular if the coefficient in front of the highest derivative term (y'') is a function that is analytic at that point. In this case, the given differential equation is:
(sin x)y'' + xy' + (x - 1/9) y = 0
The coefficient in front of y'' is sin x. Sin x is analytic for all x, so there are no singular points due to sin x.
2. Deriving the indicial equation:
Since there are no singular points due to sin x, we can skip this step.
3. Determining the form of two linearly independent solutions:
Since there are no singular points due to sin x, we can skip this step.
4. Finding the behavior of solutions as x approaches 0:
To determine the behavior of solutions as x approaches 0, we consider the term with the smallest power of x (x^0) in the equation. In this case, the term is (x - 1/9)y.
As x approaches 0, the behavior of the solutions depends on the behavior of this term. When x is close to 0, the term (x - 1/9) becomes very close to -1/9. So, the behavior of solutions as x approaches 0 is similar to the behavior of solutions of the equation y' + (-1/9)y = 0.
The solution of y' + (-1/9)y = 0 is y = Ce^(-x/9), where C is a constant. Therefore, as x approaches 0, the behavior of solutions for the given differential equation is y = Ce^(-x/9), where C is a constant.
In summary:
1. There are no regular singular points for the given differential equation.
2. Indicial equations are not required in this case.
3. Finding the form of two linearly independent solutions is not required in this case.
4. The behavior of solutions as x approaches 0 is y = Ce^(-x/9), where C is a constant.