Javascript required
Skip to content Skip to sidebar Skip to footer
Home / Find a General Solution to the Given Cauchy Euler Equation

Find a General Solution to the Given Cauchy Euler Equation

Calculus II Topics


Cauchy-Euler Equations

       Here we will learn about a specific type of second-order differential equation. If we have a differential equation of the form:

Cauchy-Euler Equation Form


it is a Cauchy-Euler equation. To solve these, we plug the coefficients into the formula:

Cauchy-Euler Formula


This gives the characteristic equation. From there, we solve for m. In a Cauchy-Euler equation, there will always be 2 solutions, m 1 and m 2; from these, we can get three different cases. Be sure not to confuse them with a standard higher-order differential equation, as the answers are slightly different. Here they are, along with the solutions they give:
  • Distinct Real Roots:

Cauchy-Euler Distinct Real Roots

  • Complex Roots:

Cauchy-Euler Complex Roots

  • Repeated Roots:
For repeated roots, we need to do a bit more work.

Cauchy-Euler Equations Repeated Double Roots


Note that the first term never has a coefficient (other than 1). There are multiple ways to solve Cauchy-Euler equations with different coefficients, however it is easier to learn just one way (having no unique coefficient for the first term). If there is a Cauchy-Euler equation that has a unique coefficient on the first term, divide it out first to isolate it.

       Let's look at a few:

Example 1: Distinct Real Roots

       Solve the following Cauchy-Euler Equation:

Cauchy-Euler Distinct Real Roots

       We can see that there is no coefficient for the first term. We can also see that A = 5 and B = -21; therefore, we are ready to use the formula:

Cauchy-Euler Formula Distinct Real Roots

       Now that we have the characteristic equation, we can solve for m:

Cauchy Euler Equation Real Distinct Roots Finding m

       There are two distinct real roots, 3 and -7:

Cauchy-Euler Equation Distinct Real Roots Solution

Example 2: Complex Roots

       Solve the following Cauchy-Euler Equation:

Cauchy-Euler Equation Complex Roots

       First, we need to divide every term by 2 in order to remove the coefficient of the first term:

Cauchy-Euler Equations Complex Imaginary Roots Isolating First Term

       Now, we have the equation in the proper form. We can see that A = 5 and B = 13, so:

Cauchy-Euler Equations Complex Imaginary Roots Formula

       Next, we solve for m. This equation can't be factored, but we can complete the square:

Cauchy-Euler Equation Complex Imaginary Finding m

       These roots are complex:

Cauchy-Euler Equation Complex Imaginary Roots Solved

Example 3: Repeated Roots

       Solve the following Cauchy-Euler Equation:

Cuachy-Euler Equation Repeated Double Roots

       First, we need to get rid of the coefficient on the first term, so let's divide everything by 4:

Cauchy-Euler Equation Repeated Double Roots

       We can now see that A = 3 and B = 1:

Cauchy-Euler Equations Repeated Double Roots Formula

       Now, we can solve for m:

Cauchy-Euler Equation Repeated Double Roots Finding m

       We have a double root of m = -1. multiplicity 2. We will need to find y 1 and y 2; y 1 is going to be:

Cauchy-Euler Equation Solving a Repeated Double Root

       Now, to find y 2, we will need to do a bit more work:

Cauchy-Euler Equation Solving a Repeated Double Root

       First, we need to start off by finding U. Remember, y 1 = x -1 and A = 3:

Cauchy-Euler Equation Solving a Repeated Double Root

       Next, we can move on to the next part of the equation:

Cauchy-Euler Equation Solving a Repeated Double Root

       We just found U; integrating to find u gives us:

Cauchy-Euler Equations Solving a Repeated Double Root

       Now, we just multiply u and y 1 to get y 2:

Cauchy-Euler Equation Solving a Repeated Double Root

       Finally, we add y 1 and y 2 to get y, remembering to include the constants:

Cauchy-Euler Equations Repeated Double Root Solved


Hello! My name is Alex, and I am the creator of CopingWithCalculus.com. I'm not a genius or a math guru; in fact, I struggled with it for several years before becoming proficient. Eventually, I passed the AP Calculus exam in high school, and Calculus II and III in college; because I struggled with it, I understand how necessary clear, concise explanations are. That is why I created this site, and am working on a book to go along with it - to help you cope with calculus!
If you have any questions or comments, don't hesitate to send an email
Copyright � 2013 Alex Makar

Find a General Solution to the Given Cauchy Euler Equation

Source: https://copingwithcalculus.com/cauchy-euler-equations.html