Consider ∂/∂t f(tx, ty). Prove that if f is homogenous of degree n, then the attatched equation is true.
Consider ∂/∂t f(tx, ty). Prove that if f is homogenous of degree n, then the attatched equation is true.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider ∂/∂t f(tx, ty). Prove that if f is homogenous of degree n, then the attatched equation is true.
![The equation shown is:
\[ x f_x(x, y) + y f_y(x, y) = n f(x, y) \]
This equation represents a form of Euler's homogeneous function theorem in partial differential equations. It states that if \( f(x, y) \) is a homogeneous function of degree \( n \), then it satisfies the above partial differential equation. In this context:
- \( f(x, y) \): A function of two variables, \( x \) and \( y \).
- \( f_x(x, y) \) and \( f_y(x, y) \): Partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.
- \( n \): The degree of homogeneity of the function \( f(x, y) \).
This equation is useful in mathematical fields where scaling properties and symmetries are analyzed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5de9-a3ed-408a-b8ea-03a65e9854b7%2F03a5cb5d-1af9-4f8d-b8f3-ef1e7e9f015c%2F5yx46h_processed.png&w=3840&q=75)
Transcribed Image Text:The equation shown is:
\[ x f_x(x, y) + y f_y(x, y) = n f(x, y) \]
This equation represents a form of Euler's homogeneous function theorem in partial differential equations. It states that if \( f(x, y) \) is a homogeneous function of degree \( n \), then it satisfies the above partial differential equation. In this context:
- \( f(x, y) \): A function of two variables, \( x \) and \( y \).
- \( f_x(x, y) \) and \( f_y(x, y) \): Partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.
- \( n \): The degree of homogeneity of the function \( f(x, y) \).
This equation is useful in mathematical fields where scaling properties and symmetries are analyzed.
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