Solving a Second-Order Linear Ordinary Differential Equation with a Cauchy-Euler Form

Introduction

In mathematics and physics, solving differential equations is a fundamental task. This article focuses on solving the following second-order linear ordinary differential equation with a Cauchy-Euler form:

[(x^2frac{d^2y}{dx^2} - xfrac{dy}{dx} 1)y 0)]

The objective is to provide a detailed solution, utilizing the method of characteristic equations and substitution of variable forms, to determine the general solution for this specific differential equation.

Step 1: Rewrite the Differential Equation in Standard Form

To solve the equation, we first rewrite it in standard form:

[(frac{d^2y}{dx^2} - frac{1}{x}frac{dy}{dx} - frac{1}{x^2} 0)]

Step 2: Assume a Specific Form for y

The given equation resembles a Cauchy-Euler equation. A common approach is to assume a solution of the form y x^m. This assumption simplifies the equation significantly.

[(y x^m)]

Then, we compute the first and second derivatives:

[(frac{dy}{dx} mx^{m-1})]

[(frac{d^2y}{dx^2} m(m-1)x^{m-2})]

Step 3: Substitute y and its Derivatives into the Equation

Substitute these into the differential equation:

[(x^2[m(m-1)x^{m-2}] - x[mx^{m-1}] 1 0)]

Simplify this expression:

[(m(m-1)x^m - mx^m 1 0)]

Factor out x^m:

[(m^2 - m - 1 0)]

Step 4: Solve the Characteristic Equation

Since we need the left-hand side to equal zero, we set:

[(m^2 - m - 1 0)]

This is a quadratic equation. Solving for m:

[(m frac{1 pm sqrt{1 4}}{2} frac{1 pm sqrt{5}}{2})]

The roots are complex:

[(m frac{1 sqrt{5}}{2}, quad m frac{1 - sqrt{5}}{2})]

Step 5: General Solution Using Complex Roots

The general solution for a Cauchy-Euler equation with complex roots can be expressed as:

[(y C_1x^{{text{Re}(m)}}cos(text{Im}(m)ln(x)) C_2x^{{text{Re}(m)}}sin(text{Im}(m)ln(x)))]

Since text{Re}(m) 0 and text{Im}(m) frac{sqrt{5}}{2}, the solution simplifies to:

[(y C_1cos(frac{sqrt{5}}{2}ln(x)) C_2sin(frac{sqrt{5}}{2}ln(x)))]

where C_1 and C_2 are arbitrary constants.

Conclusion

The solution to the differential equation is: