Cauchy-Euler Equation Solved Problems

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

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.