\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Differential Equations - Cauchy-Euler Equation

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

A Cauchy-Euler Differential Equation (also called Euler-Cauchy Equation or just Euler Equation) is an equation with polynomial coefficients of the form \(\displaystyle{ t^2y'' +aty' + by = 0 }\).
These may seem kind of specialized, and they are, but equations of this form show up so often that special techniques for solving them have been developed. We will look at a couple of techniques on this page and direct you to other techniques that will also work on other pages.

this page

 

other pages

substitution

 

reduction of order

trial solution

 

integrating factors

Technique 1 - - Substitution

A good substitution for this type of equation is \(y=\ln(t)\). This substitution converts the differential equation into one with constant coefficients.

Technique 2 - - Trial Solution

A second technique involves using a trial solution \(y=t^k\) where k is a constant. This video takes you through a general equation. The equations developed in this video are used in several practice problems below.

Houston Math Prep - Cauchy-Euler Differential Equations (2nd Order) [12mins-56secs]

Okay, before we go on to another type of differential equation with polynomial coefficients, let's work some practice problems.

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Practice

Unless otherwise instructed, solve these Cauchy-Euler differential equations using the techniques on this page.

\(\displaystyle{ x^2u'' + 3xu' + \frac{5u}{4} = 0 }\)

Problem Statement

\(\displaystyle{ x^2u'' + 3xu' + \frac{5u}{4} = 0 }\)

Final Answer

\(\displaystyle{ u = x^{-1} \left[ A\cos((1/2)\ln(x)) + B\sin((1/2)\ln(x)) \right] }\)

Problem Statement

\(\displaystyle{ x^2u'' + 3xu' + \frac{5u}{4} = 0 }\)

Solution

620 video solution

Final Answer

\(\displaystyle{ u = x^{-1} \left[ A\cos((1/2)\ln(x)) + B\sin((1/2)\ln(x)) \right] }\)

Log in to rate this practice problem and to see it's current rating.

\( 2t^2y'' + 3ty' + 5y = 0 \)

Problem Statement

\( 2t^2y'' + 3ty' + 5y = 0 \)

Solution

The Lazy Engineer - 2228 video solution

Log in to rate this practice problem and to see it's current rating.

\( t^2y'' + 5ty' + 4y = 0 \)

Problem Statement

\( t^2y'' + 5ty' + 4y = 0 \)

Solution

The Lazy Engineer - 2229 video solution

Log in to rate this practice problem and to see it's current rating.

\( x^2y'' + 7xy' + 8y = 0 \)

Problem Statement

\( x^2y'' + 7xy' + 8y = 0 \)

Solution

Houston Math Prep - 2230 video solution

Log in to rate this practice problem and to see it's current rating.

\( 9x^2y'' + 3xy' + y = 0 \)

Problem Statement

\( 9x^2y'' + 3xy' + y = 0 \)

Solution

Houston Math Prep - 2231 video solution

Log in to rate this practice problem and to see it's current rating.

\( x^2y'' - 9xy' + 28y = 0 \)

Problem Statement

\( x^2y'' - 9xy' + 28y = 0 \)

Solution

Houston Math Prep - 2232 video solution

Log in to rate this practice problem and to see it's current rating.

\( x^2 y'' - 11xy'+85y = 0, y(1) = -3, y'(1) = -4 \)

Problem Statement

\( x^2 y'' - 11xy'+85y = 0, y(1) = -3, y'(1) = -4 \)

Solution

MIP4U - 2291 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Differential Equations

Log in to rate this page and to see it's current rating.

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

effective study techniques

Shop Amazon - New Textbooks - Save up to 40%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Practice Search

Practice Instructions

Unless otherwise instructed, solve these Cauchy-Euler differential equations using the techniques on this page.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics