How to Solve the Cauchy-Euler Differential Equation?

Cauchy-Euler differential equations require a few systematic steps to solve as explained below:

Identification: First, you must verify that the differential equation is of the Cauchy-Euler form [Tex]ax^2y’’ + bxy’ + cy = 0[/Tex], where the coefficients are powers of x that matches the order of the derivative they multiply.

Assumption: You then assume a solution of the form y = x^r, where r is a constant that will be determined.

Calculate Derivatives: You compute the derivatives of the assumed solution y’ and y’’.

Substitution: You have to substitute y, y’, and y’’ back into the original differential equation.

Solve for r: After substitution, you’ll get a polynomial in x whose coefficients depend on r. Now, Set this polynomial equal to zero and solve for r to find the characteristic equation.

Determining General Solution: Depending on the roots of the characteristic equation, you’ll have different forms for the general solution:

• If the roots r₁ and r₂ are real and distinct, the general solution is [Tex]y(x) = c_1x^{r_1} + c_2x^{r_2}[/Tex].
• If there is a repeated root r, the general solution is [Tex]y(x) = c_1x^r + c_2x^r\ln(x)[/Tex].
• If the roots are complex, say [Tex]r = \alpha \pm \beta i[/Tex], the general solution is [Tex]y(x) = x^\alpha(c_1\cos(\beta \ln(x)) + c_2\sin(\beta \ln(x)))[/Tex].

Applying Initial/Boundary Conditions: If you have initial or boundary conditions, you can use them to solve for the constants [Tex]c_1[/Tex] and [Tex]c_2[/Tex].

Cauchy-Euler equation, also known as the Euler-Cauchy equation, is a type of linear differential equation with variable coefficients. It has the general form [Tex] x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y’ + a_0 y = 0[/Tex]. It’s named after two famous mathematicians, Cauchy and Euler. This equation is special because it helps us understand how things change over time or space. It’s like a key that unlocks the secrets of many natural processes, like how objects move or how electricity flows.

In this article, we’ll break down what the Cauchy-Euler equation is all about, how to solve it, and where we can see it in action in the real world.