Define ordinary point and singular point of a second order linear differential equation with variable coefficients.

Introduction

A second order linear differential equation with variable coefficients is a differential equation of the form:

$$a(x)y'' + b(x)y' + c(x)y = 0$$

where $a(x)$, $b(x)$, and $c(x)$ are functions of $x$, and $y''$, $y'$, and $y$ are the second derivative, first derivative, and the function itself respectively. The terms 'ordinary point' and 'singular point' refer to the characteristics of the points $x$ in the domain of the differential equation.

Ordinary Point

An ordinary point of a differential equation is a point where all the coefficients of the equation are analytic. In other words, at an ordinary point, the functions $a(x)$, $b(x)$, and $c(x)$ are all well-defined, differentiable, and their power series representation converges to the function in some neighborhood of the point.

For instance, consider the differential equation:

$$x^2y'' - xy' + y = 0$$

Here, all points except $x = 0$ are ordinary points because the coefficients $x^2$, $-x$, and $1$ are all analytic everywhere except at $x = 0$.

Singular Point

A singular point of a differential equation, on the other hand, is a point where at least one of the coefficients of the equation is not analytic. This means that at a singular point, at least one of the functions $a(x)$, $b(x)$, or $c(x)$ is either not defined, not differentiable, or its power series does not converge to the function in any neighborhood of the point.

For the differential equation given above:

$$x^2y'' - xy' + y = 0$$

The point $x = 0$ is a singular point because the coefficients $x^2$ and $-x$ are not analytic at $x = 0$.

Comparison between Ordinary Point and Singular Point

Conclusion

Understanding the concepts of ordinary points and singular points is crucial in solving second order linear differential equations with variable coefficients. These points determine the behavior of the solution and can significantly affect the method of solving the differential equation. Therefore, identifying whether a point is ordinary or singular is a fundamental step in the process of solving such equations.

Diagram Requirement

No diagram is required for this question as it is purely theoretical and does not involve any graphical representation.

Summary

A second order linear differential equation with variable coefficients can have ordinary points and singular points. An ordinary point is a point where all the coefficients of the equation are analytic, while a singular point is a point where at least one of the coefficients is not analytic. Ordinary points have well-defined, differentiable coefficients, while singular points have coefficients that are either not defined, not differentiable, or their power series does not converge to the function in any neighborhood of the point.

Analogy

Imagine a second order linear differential equation as a road trip. The coefficients of the equation represent the condition of the road. At ordinary points, the road is smooth and well-maintained, allowing for a comfortable journey. However, at singular points, the road becomes bumpy, with potholes and obstacles, making the journey challenging and unpredictable.