Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x^{2} y^{\prime \prime}-3(\sin x) y^{\prime}+\left(1+x^{2}\right) y=0\)

Short answer

Question: Identify the singular points of the given differential equation \(x^{2} y^{\prime \prime}-3(\sin x) y^{\prime}+\left(1+x^{2}\right) y=0\) and determine if they are regular or irregular. Answer: The given differential equation has a singular point at \(x = 0\), and this singular point is regular.

Step-by-step solution

Identify the coefficients

We are given a second-order linear ODE with the following coefficients: \\ \(A(x) = x^2\) \\ \(B(x) = -3\sin(x)\) \\ \(C(x) = 1+x^2\)

Find the singular points

To find the singular points, we should check when the coefficients \(A(x)\), \(B(x)\), and \(C(x)\) are not analytic. Observe that \(A(x) = x^2\) has a singularity at \(x = 0\), since the roots of a polynomial of order 2 are not continuous at the origin. Therefore, the singular point is \(x = 0\).

Determine the nature of the singular point

To determine if the singular point \(x = 0\) is regular or irregular, we should check whether \(xB(x)\) and \(x^2 C(x)\) are analytic at the singular point: \\\(xB(x) = -3x\sin(x) \\\) \\\(x^2 C(x) = x^2 (1+x^2)\\\) Notice that both \(xB(x)\) and \(x^2C(x)\) are polynomial functions, therefore they are analytic at every point (including \(x = 0\)). From this observation, it follows that the singular point \(x = 0\) is a regular singular point.

Conclusion

The given differential equation \(x^{2} y^{\prime \prime}-3(\sin x) y^{\prime}+\left(1+x^{2}\right) y=0\) has a singular point at \(x = 0\), and this singular point is regular.

Key concepts

Second-Order Linear ODE

Second-order linear ordinary differential equations (ODEs) are a fundamental class of differential equations characterized by the presence of the second derivative of the unknown function. They have the general form \( a(x)y'' + b(x)y' + c(x)y = d(x), \) where \( y'' \) represents the second derivative of \( y \) with respect to \( x \) and \( a(x), b(x), c(x), \) and \( d(x) \) are given functions of \( x \) that determine the coefficients of the equation. These types of ODEs are essential in modeling dynamics where acceleration (the second derivative) is proportional to the position or the force acting on a body, such as in mechanical and electrical systems.

When the function \( d(x) = 0 \) the equation is called homogeneous. Our exercise deals with such a homogeneous equation, which has solutions that can be superimposed to form more complex solutions. Understanding the nature and solutions of second-order linear ODEs is critical for students in mathematics, engineering, and the physical sciences.

Regular Singular Point

Within the study of differential equations, a singular point is worth special attention, particularly when it comes to second-order linear ODEs. A singular point of a differential equation is a value of \( x \) where the coefficients do not all remain finite and analytical. However, not all singularities behave the same way. In contrast, a regular singular point is a type of singularity where the equation behaves