Virtual Lessons: Cauchy-Euler

Cauchy-Euler differential equations, also known as homogeneous linear differential equations, are a type of differential equation that can be written in the form of 

\(a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y’ + a_0(x)y = 0\), where \(a_k(x)\)

are continuous functions of \(x\) and \(y^{(n)}\) represents the \(n\)-th derivative of \(y\) with respect to \(x\). These equations have a special property where the coefficients can be expressed as powers of \(x\), which makes them solvable using a simple substitution. In this way, solving Cauchy-Euler differential equations involves converting the equation into a constant coefficient equation through a change of variable, followed by finding the general solution using standard methods such as finding the roots of the characteristic equation or using exponential functions. In this section, we will explore the techniques used to solve Cauchy-Euler differential equations and provide examples to illustrate the process.