Cauchy-Euler equations are a special type of linear ODE with variable coefficients. They're important because they pop up in many real-world problems, like vibrating systems and certain physical phenomena.

Solving these equations involves a clever substitution trick that turns them into constant coefficient ODEs. This connects to earlier methods we've learned, showing how different techniques can work together to solve more complex problems.

Cauchy-Euler Equations

Definition and Characteristics

• Cauchy-Euler equations are a special type of linear ordinary differential equation
• Have the form $ax^2y'' + bxy' + cy = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• Also known as equidimensional equations because all terms in the equation have the same dimension (sum of the powers of $x$ and the order of the derivative is the same for each term)
• Occur in various applications, such as in the study of vibrating systems and in the analysis of certain physical phenomena

Solution Approach

• The substitution method is used to solve Cauchy-Euler equations
• Substitute $x = e^t$ or $x = \ln t$, which transforms the equation into a linear differential equation with constant coefficients
• After substitution, the resulting equation can be solved using standard techniques for linear differential equations (characteristic equation, undetermined coefficients, variation of parameters)
• The final solution is obtained by replacing the substituted variable with the original variable $x$

Solving Cauchy-Euler Equations

Indicial Equation and Regular Singular Points

• The indicial equation is derived from the Cauchy-Euler equation and is used to determine the form of the solution
• For the Cauchy-Euler equation $ax^2y'' + bxy' + cy = 0$, the indicial equation is $ar(r-1) + br + c = 0$, where $r$ is the unknown
• The roots of the indicial equation determine the form of the solution (polynomial, exponential, or logarithmic)
• Cauchy-Euler equations have a regular singular point at $x = 0$, which means the solution can be represented by a power series in the neighborhood of $x = 0$

Power Series Solution and Method of Frobenius

• The power series solution assumes a solution of the form $y = \sum_{n=0}^{\infty} a_n x^{r+n}$, where $r$ is a root of the indicial equation and $a_n$ are the coefficients to be determined
• Substitute the power series into the Cauchy-Euler equation and equate coefficients of like powers of $x$ to obtain a recurrence relation for the coefficients $a_n$
• The method of Frobenius is a systematic approach to finding power series solutions for differential equations with regular singular points, such as Cauchy-Euler equations
• Frobenius method involves substituting the power series solution into the equation, determining the indicial equation, finding the recurrence relation for the coefficients, and constructing the final solution based on the roots of the indicial equation