How to Solve Cauchy Euler Differential Equations

TL;DR

To solve Cauchy Euler differential equations, start by assuming a solution of the form y = x^m. Differentiate to find derivatives, plug them into the equation, and solve the resulting quadratic for m. The solution's form will depend on the nature of the roots, which can be distinct real, repeated real, or complex conjugates.

Transcript

in this video we're going to talk about Hoshi Euler equations some Cauchy Euler equations so here's an example of a Koshi order equation x squared y double prime plus 3 XY prime plus 2 y equals 0 so the reason this is a Cauchy Euler differential equation is because the power of X matches the order of the derivative see here the power of X is 2 here... Read More

Key Insights

• ☺️ Cauchy Euler equations match the power of X with the order of the derivative for specific solutions.
• 🫚 The steps to solve Cauchy Euler equations involve making assumptions, differentiation, plugging into the equation, and solving for the roots.
• 🫚 Solutions to these equations can include distinct real roots, repeated real roots with Ln X, or complex conjugates with trigonometric functions.
• ☺️ Understanding the relationship between X power and derivative order is crucial in identifying Cauchy Euler equations.
• 🔌 Step by step solutions to Cauchy Euler equations involve careful differentiation, plugging in variables, and solving for constants.
• 🍉 Complex conjugates in Cauchy Euler equation solutions involve introducing Alpha and Beta terms with trigonometric functions.
• 🫚 Multiplicity of roots in Cauchy Euler equations affects the form of the solution, such as repeated roots requiring Ln X terms.

Questions & Answers

Q: What is the key characteristic of a Cauchy Euler differential equation?

The key characteristic is that the power of X matches the order of the derivative in the equation, distinguishing it from other types of differentials equations.

Q: How do you start solving a Cauchy Euler equation step by step?

The first step is to assume a solution of the form Y = X^M, followed by differentiation, plugging into the differential equation, and solving for the constants involved.

Q: What are the possible outcomes when solving a Cauchy Euler equation?

The outcomes can include distinct real roots leading to distinct powers of X, repeated real roots requiring multiplication by Ln X, or complex conjugates represented by trigonometric functions and X to the power of Alpha.

Q: How do you handle a repeated real root in a Cauchy Euler equation solution?

In the case of a repeated real root, the solution involves adding Ln X to the repeated term to account for the multiplicity of that root in the equation.

Summary & Key Takeaways

• Cauchy Euler equations involve matching the power of X with the order of the derivative in differential equations.

• Steps to solve Cauchy Euler equations involve assumptions, differentiation, plugging in, and solving a quadratic equation.

• Solutions can include distinct real roots, repeated real roots, or complex conjugates depending on the equation.