User:Kongrui875/Report3

=R*3.12=

Problem 12: Derive 2nd exactness condition of general N2-ODE and verify the 2nd exactness condition of a particular N2-ODE
Report problem 3.12 from.

Given: 2nd exactness condition of N2-ODE
A N2-ODE

is exact if it satisfy that a function

exists that

($$) has the form of

where

The 2nd exactness condition for N2-ODE ($$) is

A particular N2-ODE is given as

Find: Derive and verify
1.Derive ($$)

2.Derive ($$)

3.Verify ($$) satisfies the 2nd exactness condition.

Solution

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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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Since

from ($$),

take partial derivative wrt p,

from ($$) and ($$)

thus,

from ($$), take partial derivative wrt x,

substitute ($$) and ($$) into ($$),

rearrange ($$),

take partial derivative wrt p,

substitute into ($$) to find

This proved the 2nd relation in the 2nd exactness condition.

substitute ($$) into ($$),

take partial derivative wrt y,

take partial derivative of ($$) wrt y,

substitute ($$) and ($$) into ($$) to find

This proved the 1st relation in the 2nd exactness condition.

In ($$),

To verify the 1st relation in the 2nd exactness condition:

take partial derivative of ($$) wrt x,

take partial derivative of ($$) wrt x,

take partial derivative of ($$) wrt y,

take partial derivative of ($$) wrt y,

take partial derivative of ($$) wrt y,

substitute ($$), ($$) and ($$) into LHS of ($$),

take partial derivative of ($$) wrt x,

take partial derivative of ($$) wrt p,

take partial derivative of ($$) wrt y,

take partial derivative of ($$) wrt p,

substitute ($$), ($$) and ($$) into RHS of ($$),

thus,

This verified the 1st relation in the 2nd exactness condition.

To verify the 2nd relation in the 2nd exactness condition:

take partial derivative of ($$) wrt p,

take partial derivative of ($$) wrt p,

substitute ($$), ($$) and ($$) into LHS of ($$),

take partial derivative of ($$) wrt p,

take partial derivative of ($$) wrt p,

substitute ($$) into RHS of ($$),

thus,

This verified the 2nd relation in the 2nd exactness condition.

Thus, ($$) satisfies the 2nd exactness condition.