Cauchy Euler Equation

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.

Cauchy-Euler equation, also known as the Euler-Cauchy equation, is a type of ordinary differential equation (ODE) with variable coefficients that are powers of the independent variable.

It’s a linear homogeneous differential equation and is notable for being solvable in terms of power functions. Solutions to the Cauchy-Euler equation can often be found using methods such as substitution, reduction of order, or power series.

The differential equation,

[Tex]\bold{a_nx^ny^{(n)}+a_{n-1}x^{n-1}y^{n-1}+…+a_0y}[/Tex]

is called the Cauchy Euler differential equation of order n.

Where yⁿ represents the n^th derivative of y with respect to x, a_i, i = 0, . . . , n are constants and a_n ≠ 0.

Cauchy-Euler Equation Examples

Here are some examples of Cauchy-Euler equations:

• x²y” +y=0
• x²y” + 4y=0
• x²y” -3xy’ – 7y =0
• x³y” – 2x²y’ + xy =0

The most common Cauchy–Euler equation is the second-order equation, appearing in a number of physics and engineering applications, such as when solving Laplace’s equation in polar coordinates. It is given by the equation:

[Tex]\bold{a_2x^{2}{\frac {d^{2}y}{dx^{2}}}+a_1x{\frac {dy}{dx}}+a_0y=0}[/Tex]

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].

Problem 1: Solve a Cauchy-Euler equation step by step. Consider the second-order Cauchy-Euler equation:

x²y′′−6xy′+13y=0

Solution:

Assume a solution of the form [Tex]y = x^r[/Tex].

The derivatives are [Tex]y’ = rx^{r-1} and y’’ = r(r-1)x^{r-2}[/Tex].

Substitute y, y’ , and y’’ back into the original equation:

[Tex]x^2(r(r−1)x^{r−2})−6x(rx^{r−1})+13x^r=0[/Tex]

This simplifies to:

[Tex]r(r−1)x^r−6rx^r+13x^r=0[/Tex]

Factor out [Tex]x^r[/Tex] [Tex]since ( x \neq 0 )[/Tex] and solve the characteristic equation:

r(r−1)−6r+13=0

⇒ [Tex]r^2−7r+13=0[/Tex]

This is a quadratic equation in r .

Since the discriminant b^2 – 4ac is negative the roots of the characteristic equation are complex . Let’s find the roots:

[Tex]r=\dfrac{7±\sqrt{49−4(1)(13)}}{2}[/Tex]

⇒ [Tex]r=\dfrac{7±\sqrt{49−52}}{2}[/Tex]

⇒ [Tex]r=\dfrac{7±\sqrt{-3}}{2}[/Tex]

⇒ [Tex]r=\dfrac{7}2±\dfrac{\sqrt3i}2[/Tex]

The roots are complex hence, the general solution is:

[Tex]y(x)=x^{7/2}(c_1cos(\dfrac{\sqrt3}2ln(x))+c_2sin(\dfrac{\sqrt3}2ln(x)))[/Tex]

Here, [Tex]c_1[/Tex] and [Tex]c_2[/Tex] are constants determined by boundary conditions or initial values.

Problem 2: Solve Cauchy-Euler equation step by step. Consider the second-order Cauchy-Euler equation:

x²y’’ – 7xy’ + 16y = 0

Solution:

Let’s assume that y = x^ris the solution of the given differential equation, where [Tex](r)[/Tex] is a constant to be determined.

Substitute y = x^r into the differential equation:[Tex][ x^2y’’ – 7xy’ + 16y = 0] [x^2[r(r-1)x^{r-2}] – 7x[rx^{r-1}] + 16x^r = 0][/Tex]

For the first derivative term: [Tex]x^2[r(r-1)x^{r-2}] = r(r-1)x^r[/Tex]

For the second derivative term: [Tex]7x[rx^{r-1}] = 7rx^r[/Tex]

For the third term: 16x^r

Combining all terms: [Tex]r(r-1)x^r – 7rx^r + 16x^r = 0[/Tex]

Set the polynomial equation equal to zero: r(r-1) – 7r + 16 = 0

Solving this quadratic equation for r: [Tex]r^2 – 8r + 16 = 0[/Tex]

Factoring: [Tex](r-4)^2 = 0[/Tex] The repeated root is r = 4.

Since we have a repeated root, the general solution is: [Tex]y(x) = c_1x^4 + c_2x^4\ln(x)[/Tex]

Also Read,

• Euler Method for solving differential equation
• Cauchy’s Mean Value Theorem
• Euler’s Theorem

What is a Cauchy-Euler Equation?

A Cauchy-Euler equation, also known as an Euler-Cauchy equation or Euler’s equation, is a linear, homogeneous, ordinary differential equation with variable coefficients.

Write general form of Cauchy-Euler Equation.

It is generally of the form:

[Tex]a_nx^ny^{(n)}+a_{n-1}x^{n-1}y^{n-1}+…+a_0y[/Tex]

Why are they called Cauchy Euler equations?

They are named after the mathematicians Augustin-Louis Cauchy and Leonhard Euler who made significant contributions to the theory of differential equations.

Can Cauchy-Euler Equations have non-homogeneous forms?

Yes, Cauchy-Euler equations can have non-homogeneous forms. A second order Euler-Cauchy differential equation is non-homogeneous if g(x) is non-zero. For example, the equation [Tex]x^{2}y”-2y=x^{3}e^{x}[/Tex] is a non-homogeneous 2nd order Euler-Cauchy differential equation.