Corollary 3 of Euler's Theorem Formula and Proof  Summary and Q&A
TL;DR
Corollary 3 of Euler's Theorem states that if a function u in x and y with degree n has a function z in terms of u that is homogeneous in x and y with degree n, then x d z by d x plus y d z by d y will equal n times z.
Questions & Answers
Q: What does Corollary 3 of Euler's Theorem state?
Corollary 3 states that if there is a function u in x and y with degree n and a function z in terms of u that is homogeneous with degree n in x and y, then x d z by d x plus y d z by d y should be equal to n times of z.
Q: How is Corollary 3 related to Corollary 2?
Corollary 2 states that for a function z in terms of u where u may not be homogeneous, we have the equation x times of d u over d x plus y times of d u over d y equal to n times of f of u over f dash of u. Corollary 3 can be derived from Corollary 2 by assuming n times of f of u over f dash of u as g of u.
Q: What is the proof for Corollary 3?
The proof involves differentiating two key equations. The first equation is obtained by applying the product rule to differentiate x d z by d x plus y d z by d y. The second equation is obtained by differentiating the first equation with respect to y. By combining and simplifying these equations, the result provided by Corollary 3 is obtained.
Q: What happens when the partial differential equation is simplified?
Simplifying the partial differential equation leads to x square d 2 u by d x square plus 2 x y d 2 u by d x d y plus y square d 2 u by d y square equal to g dash u minus 1 times of g u, which is the result of Corollary 3.
Summary & Key Takeaways

Corollary 3 of Euler's Theorem states that if there is a function u in x and y with degree n and a function z in terms of u that is homogeneous with degree n in x and y, then a certain partial differential equation holds.

The proof for Corollary 3 involves differentiating two key equations, one obtained from applying the product rule and the other obtained from differentiating the first equation with respect to y.

Finally, combining these equations and simplifying leads to the result provided by Corollary 3 of Euler's Theorem.