User:Cloudwu890202/Report 5.8

=R*5.8=

Problem 8: Show the 2nd Exactness Condition
Report problem 5.8 from lecture notes section 22-6.

Given: General Equation of N2-ODE
One general Equation of N2-ODE with the form of 1st exactness condition shown as below

which can be also expressed as below

where

Define

Thus Eq.(8.2) can be rewritten as

Find: show the 2nd Exactness Condition
Since $$1$$ and $$q$$ are lineraly independent ($$q=y''$$ is in general not a constant), we must have

In other words, show the 2nd Exactness about N2-ODE

Solution: work with coefficients in 2nd exactness condition

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

where

In Eq.(8.1), $$g$$ and $$f$$ are all functions of $$(x,y,y')$$ that are considered as 3 independent variables. Thus $$ g_i$$ for $$ i=0,1,2 $$ can be expanded as

Next

Thus, Eq.(8.8) can be expanded as

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

This is 2nd Exactness Condition