Homogeneous function of degree n

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Homogeneous functions of degree n are mathematical functions that exhibit a specific scaling property, where scaling the input by a factor results in the output being scaled by a power of that factor. The discussion revolves around understanding these functions in the context of partial differential equations. The user has successfully solved part a of the homework but is struggling with part b, which requires proving that the mixed partial derivatives f_{xy} and f_{yx} are equal. There is a concern that the problem statement lacks sufficient equations to complete this proof. Clarification on the properties of homogeneous functions and their derivatives is needed to assist in solving part b.
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Homework Statement


See photo


Homework Equations





The Attempt at a Solution


I am learning partial differential and never be taught about homongeneous function.
What is this? How to solve the problem?
 

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I finally figure out how to do part a.
See my work on the photo.

Part b is similar to part a except 1 step.
That is, I need to prove f_{xy}=f_{yx}

It seems the question doesn't provide enough equation for me to do this.
 

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Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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