User:Cloudwu890202/Report 5.7

=R*5.7=

Problem 7: Show the Equivalence
Report problem 5.7 from lecture notes section 22-4.

Given: 2 forms of 2nd exactness condition of N2-ODE
One form of 2nd exactness condition of N2-ODE shown as below

where

The other form of 2nd exactness condition of N2-ODE shown as below

where

Find: show the equivalence of these two forms
Show this equivalence to symmetry of mixed 2nd partial derivatives of first integral $$\phi$$

Solution: expand 1st form to show that it is equal to the 2nd form

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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As the definition

The first integral $$\phi$$ is a function of $$(x,y,y')$$ that are considered as 3 independent variables. Thus $$ g_i$$for $$ i=0,1,2 $$ can be expanded as

To make it easier to get the solution, I just expand the derivatives not too far toward the detal(bottom) level; instead, just stay at the more general(top) level as much as possible as shown below.

Thus, Eq.(7.1) can be expanded as

p and q are not generally equal to 0, thus in order to make the Eqn to be equal to 0, the terms of this Eqn must be satiesfied for these conditions as below

Which means