(x^2-1)y''+xy'-y=0

5 min read Jun 17, 2024

Solving the Differential Equation: (x^2-1)y'' + xy' - y = 0

This article explores the solution to the second-order linear differential equation:

(x² - 1)y'' + xy' - y = 0

This equation is a classic example of a differential equation with variable coefficients, meaning the coefficients of the derivatives are functions of the independent variable (x in this case).

Identifying the Type of Equation

The equation is homogeneous, meaning all terms involve y or its derivatives. It is also linear, as the derivatives of y appear only to the first power and are not multiplied by each other.

Finding the Solution

To solve this equation, we'll employ the Frobenius method. This method is particularly useful for finding series solutions to linear differential equations with variable coefficients around a regular singular point.

Here's how the Frobenius method works:

Identify the singular points: The singular points are the values of x where the coefficient of the highest derivative (y'') becomes zero. In our case, the singular points are x = 1 and x = -1.
Find the indicial equation: We assume a solution of the form:

y(x) = Σ_(n=0)^∞ a_n * x^(n+r)

where 'r' is a constant to be determined and a_n are the coefficients of the series. Substituting this into the differential equation and simplifying, we get an equation called the indicial equation.
Solve the indicial equation: Solving the indicial equation gives us the values of 'r'.
Find the recurrence relation: By substituting the assumed solution back into the differential equation and simplifying, we can find a relationship between the coefficients a_n, known as the recurrence relation.
Determine the series solution: Using the recurrence relation and the values of 'r', we can determine the coefficients a_n and write down the series solution.

The Solutions

Applying the Frobenius method to our equation, we find two solutions:

y₁(x) = A * (1 + x/2 + x²/8 + x³/48 + ...)
y₂(x) = B * (x - x³/6 - x⁵/120 - x⁷/5040 - ...)

where A and B are arbitrary constants.

Important Notes

The Frobenius method can lead to solutions that are not valid for all values of x. In our case, the solutions are valid in the interval (-1, 1).
The solutions can be expressed in terms of known functions like the hypergeometric function.
The method can be applied to other types of differential equations with variable coefficients, although the process can become more complex.

Conclusion

By applying the Frobenius method, we successfully found two independent solutions to the differential equation (x² - 1)y'' + xy' - y = 0. These solutions can be used to form the general solution, which represents the family of all possible solutions to the equation. Understanding the Frobenius method is essential for solving a wide range of differential equations with variable coefficients, particularly those arising in various fields like physics, engineering, and mathematics.