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Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a homogeneous function with respect to t. ) Prove that 28²f of əxəy 20² f dy² = n(n-1) f(x, y). əx² +2ry-

Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a homogeneous function with respect to t. ) Prove that 28²f of əxəy 20² f dy² = n(n-1) f(x, y). əx² +2ry-

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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4. A function f is called homogeneous of degree n E N if, for all t e R, f(tx, ty) = t" f(x, y). Consider a homogeneous function of degree n that is at least twice differentiable. (a) Prove that af af ду + y =nf(x,y). (Hint: Differentiate both sides of the definition of a homogeneous function with respect to t. ) (b) Prove that + 2xy Jrdy — п(п — 1)f(г, у). + y?. ду?
Transcribed Image Text:4. A function f is called homogeneous of degree n E N if, for all t e R, f(tx, ty) = t" f(x, y). Consider a homogeneous function of degree n that is at least twice differentiable. (a) Prove that af af ду + y =nf(x,y). (Hint: Differentiate both sides of the definition of a homogeneous function with respect to t. ) (b) Prove that + 2xy Jrdy — п(п — 1)f(г, у). + y?. ду?
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