Euler-Cauchy equations

The Cauchy-Euler or Euler-Cauchy is a linear differential equation that is written in the form:

Where the coefficients $a_0,a_1,…,a_n$ are constants, and a_n ≠ 0. Let’s learn how to write the Cauchy-Euler equation and how to solve various types of Cauchy-Euler equations here in this answer.

In general, we use infinite series to solve DEsDifferential equation with non-constant coefficients. A second-order linear differential equation is written in the form:

How to solve the Cauchy-Euler equation

There are three different cases that are taken into account, depending on whether the roots of this quadratic equation are real and distinct, real and equal, or complex. In the last case, the roots are a conjugate pair. The solutions to Cauchy-Euler equations can be found using the characteristic equation:

Case 1: Distinct real roots

Let $m_1$ and $m_2$ denote the real roots of (2) such that $m_1$$\neq$$m_2$. Then $y_1 = x^{m1}$ and $y_2 = x^{m2}$ form a fundamental set of solutions. The general solution for this would be:

Example

Find the general solution of the following differential equation:

Let $y=x^m$ be the solution. This leads to:

So, the auxiliary equation is:

Our next step is to find the roots. We use the quadratic equation and then plug the roots in the general solution.

Hence:

Case 2: Repeated roots

This case solves the problem when we run into double or repeated roots. The second solution to this would be:

Hence, the general solution in this case is,

Example

Find the general solution to the following differential equation:

Let $y=x^m$ be the solution. This leads to:

So, the general solution is:

Case 3: Complex conjugate

In this case, our roots will be in the form:

Hence, the general solution would be:

Example

Let's find the general solution of the following DE:

Hence, the general solution for this is:

Wrap up

The Cauchy-Euler Equation plays an important role in the theory of linear differential equations because of its direct application to Fourier's method in deconstructing PDEs (partial differential equations).