Mathematical Step-by-Step Derivation

This expression is a standard second-order homogeneous Cauchy-Euler differential equation, which takes the form:

$$x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} + 2y = 0$$

To solve a Cauchy-Euler equation, we assume a solution of the form $y = x^m$. Finding its derivatives yields:

$\frac{dy}{dx} = m x^{m-1} \implies x \frac{dy}{dx} = m x^m$

$\frac{d^2y}{dx^2} = m(m-1) x^{m-2} \implies x^2 \frac{d^2y}{dx^2} = m(m-1) x^m$

Substituting these values back into the primary differential equation gives:

$$[m(m-1) + 4(m) + 2]\, x^m = 0$$

Since $x^m \neq 0$, we can solve for the auxiliary characteristic equation:

$$m^2 - m + 4m + 2 = 0$$

$$m^2 + 3m + 2 = 0$$

Factoring the quadratic polynomial:

$$(m+1)(m+2) = 0 \implies m_1 = -1, \;\; m_2 = -2$$

Substituting the roots back into our assumed format yields the general solution:

$$y = c_1 x^{-1} + c_2 x^{-2} \implies y = \frac{c_1}{x} + \frac{c_2}{x^2}$$