1. The problem is to write down the Cauchy-Euler differential equation. 2. The Cauchy-Euler differential equation is a type of linear differential equation with variable coefficients that can be written in the form: $$x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y' + a_0 y = 0$$ where $y^{(k)}$ denotes the $k$-th derivative of $y$ with respect to $x$, and $a_0, a_1, \ldots, a_{n-1}$ are constants. 3. For the second-order case, which is the most common, the equation is: $$x^2 y'' + a x y' + b y = 0$$ where $a$ and $b$ are constants. 4. Important rules: - The equation has variable coefficients that are powers of $x$ matching the order of the derivative. - It is often solved by substituting $y = x^m$ and finding the characteristic equation. 5. This form is useful in many applications such as physics and engineering where the coefficients depend on the independent variable in a power-law manner. Final answer: The general Cauchy-Euler differential equation of order $n$ is: $$x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y' + a_0 y = 0$$ and the common second-order form is: $$x^2 y'' + a x y' + b y = 0$$