Solving the Differential Equation x^2 d^2y/dx^2 x dy/dx - y 0: A Comprehensive Guide

In this article, we discuss how to solve the differential equation:

x^2 frac{d^2y}{dx^2} - x frac{dy}{dx} - y 0

Introduction to the Cauchy-Euler Equation

The given differential equation is known as a Cauchy-Euler equation, which is a type of second-order linear differential equation. Such equations can be written in the standard form:

a x^2 frac{d^2y}{dx^2} b x frac{dy}{dx} c y 0

In this case, our equation is:

x^2 frac{d^2y}{dx^2} - x frac{dy}{dx} - y 0

with coefficients (a 1), (b -1), and (c -1).

Solving the Differential Equation

Step 1: Assume a Solution of the Form (y x^m)

One common method to solve a Cauchy-Euler equation is to assume a solution of the form:

y x^m

Let's compute the derivatives:

frac{dy}{dx} m x^{m-1}

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

Step 2: Substitute into the Differential Equation

Now, substitute these expressions into the original differential equation:

x^2 cdot m(m-1) x^{m-2} - x cdot m x^{m-1} - x^m 0

This simplifies to:

m(m-1) x^m - m x^m - x^m 0

Factoring out (x^m) (assuming (x eq 0)), we get:

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

Step 3: Solve the Characteristic Equation

Setting the expression in brackets to zero, we obtain:

m(m-1) - m - 1 0

This can be factored as:

m^2 - m - m - 1 0

Simplifying, we get:

m^2 - 1 0

Which factors into:

(m - 1)(m 1) 0

Hence, the roots are:

m 1 quad text{and} quad m -1

Step 4: Write the General Solution

The general solution of the differential equation is a linear combination of the solutions corresponding to these roots:

y(x) C_1 x^1 C_2 x^{-1}

or simplified as:

y(x) C_1 x frac{C_2}{x}

Definition and Conceptual Understanding

Cauchy-Euler Equation

A Cauchy-Euler equation is characterized by the fact that the power of (x) (the base function) is equal to the order of the derivative. This equation is usually solved by assuming a solution of the form (y x^r) and then finding the values of (r).

The roots of the characteristic polynomial, (r_{1}, r_{2}), are derived from the simplified characteristic equation:

r^2 - 1 0

which gives:

r_{1} -1 quad text{and} quad r_{2} 1

Equidimensional Substitution

Another way to approach the equation is through the substitution method for equidimensional differential equations:

y x^r

After substituting and simplifying, we again end up with: