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WMDhamnekar
MHB
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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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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.
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.
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.
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.
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.