# Euler equations having double roots as a solution

• MHB
• WMDhamnekar
In summary, the proof for the second solution of the Euler equations with double roots is based on the variation of parameters method. By converting the equation to one with constant coefficients, it can be shown that the characteristic equation and values are the same, resulting in a general solution of Ax^r+ B\ln(x) x^r.
WMDhamnekar
MHB
If the Euler equations have double roots as it's solution, second solution will be $y_2(x)=x^r\ln{x}$. what is its proof? or how it can be derived?

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I believe variation of parameters is the usual proof method.

The change of variable, $u= \ln(x),$ converts an "Euler type equation" (also known as an "equipotential equation") to a differential equation with constant coefficients. If $$ax^2\frac{d^2y}{dx^2}+ bx\frac{dy}{dx}+ cy= 0$$ then, with $$u= \ln(x)$$, $$\frac{dy}{dx}= \frac{dy}{du}\frac{du}{dx}= \frac{1}{x}\frac{dy}{du}$$ and $$\frac{d^2y}{dx^2}= \frac{d}{dx}\left(\frac{1}{x}\frac{dy}{du}\right)= -\frac{1}{x^2}\frac{dy}{du}+ \frac{1}{x}\frac{d}{dx}\frac{dy}{du}= -\frac{1}{x^2}\frac{dy}{du}+ \frac{1}{x^2}\frac{d^2y}{du^2}$$.So $$ax^2\frac{d^2y}{dx^2}+ bx\frac{dy}{dx}+ cy= a\frac{d^2y}{du^2}- a\frac{dy}{du}+ b\frac{dy}{du}+ cy= a\frac{d^2y}{du^2}+ (b- a)\frac{dy}{du}+ cy= 0$$.

The characteristic equation for that constant-coefficients equation is the same as for the Euler-type equation so both have the same characteristic values. In particular, if the characteristic equation has a double root, r, then the constant-coefficients equation has the general solution $$y(u)= Ae^{ru}+ Bue^{ru}$$. Since $u= \ln(x)$ the general solution in terms of $x$ becomes $$y(x)= Ae^{r \ln(x)}+ B \ln(x) e^{r \ln(x)}= A e^{\ln(x^r)}+ B \ln(x) e^{\ln(x^r)}= Ax^r+ B\ln(x) x^r.$$

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## What are Euler equations?

Euler equations are a set of differential equations that describe the motion of a fluid in a two-dimensional or three-dimensional space. They are named after the Swiss mathematician Leonhard Euler.

## What are double roots in the context of Euler equations?

Double roots refer to the situation where the solution to the Euler equations is a repeated root. In other words, the equations have two identical solutions instead of two distinct solutions.

## Why is having double roots as a solution significant in Euler equations?

Having double roots as a solution can indicate the presence of a critical point in the fluid flow, where the velocity and other physical properties of the fluid may experience sudden changes. This can have important implications for understanding and predicting the behavior of fluid systems.

## What are the implications of double roots in the stability of a fluid system?

Double roots can indicate a loss of stability in a fluid system. This means that small disturbances or changes in the system can lead to large and unpredictable effects, making it difficult to control or predict the behavior of the fluid.

## How do scientists approach the study of Euler equations with double roots?

Scientists use mathematical tools and techniques, such as perturbation analysis and numerical simulations, to study the behavior of fluids described by Euler equations with double roots. They also conduct experiments to validate their findings and gain a deeper understanding of the underlying physical processes.

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