How to Solve a Fourth Order Cauchy Euler Differential Equation

TL;DR

To solve a fourth order Cauchy Euler differential equation, substitute Y with X to the power of M. Derive multiple times to find M, resulting in distinct roots. The final solution combines terms of X raised to the respective powers along with constants.

Transcript

this problem we're going to solve this differential equation so this is a Cauchy Euler differential equation the reason is if you were to multiply everything by X cubed you would get X cubed times X is X to the fourth and then here you would get plus ten X cubed Y triple prime and then you would notice that the power of X matches the order of the d... Read More

Key Insights

• ☺️ Cauchy Euler differential equations involve matching X powers to derivative orders for specific forms.
• 🤶 Solving these equations requires assuming Y as X to the M and finding the roots for M to derive the final solution.
• ✊ The final solution consists of a combination of X raised to different powers, each multiplied by constants.

Questions & Answers

Q: How can Cauchy Euler differential equations be identified?

Cauchy Euler differential equations are identified when the power of X matches the order of the derivative, leading to a specific form of differential equation that helps in solving.

Q: What is the initial step in solving a Cauchy Euler differential equation?

The initial step involves assuming Y to be equal to X to the M and finding the derivatives up to the order given in the differential equation.

Q: How are roots calculated in the process of solving Cauchy Euler differential equations?

Roots are calculated by substituting the assumed Y back into the differential equation, grouping like terms, factoring out X to the M, and solving for the roots of M.

Q: What is the significance of finding the roots in solving Cauchy Euler differential equations?

Finding the roots of M helps in constructing the final solution, which is a combination of X raised to different powers, each multiplied by constants determined by the roots.

Summary & Key Takeaways

• Cauchy Euler differential equations involve matching derivative orders to powers of X.

• Solving involves substituting Y with X to the M and finding the roots for M.

• The final solution is a combination of X raised to different powers with constants.