User:Egm6321.f12.team7.parikh.a/R3.6

Solution: Test the function with 1st and 2nd exactness condition and find the integrating factor h

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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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To examine whether the function is exact is to find out that whether the function satisfies the 1st and 2nd Exactness Condition.

1. 1st exactness condition satisfies

with

2. 2nd exactness condition could be found as below

Apparently,

Thus, This function satisfying the 1st exactness condition, however, not satisfying the 2nd exactness condition.

To make the function exact by the IFM, first find the integrating factor h(x,y) such that the following function can be exact

It is given that

Then find the n(x) as below

The integrating factor h(x) can be calculate as below

If the $$ d_1\neq 0$$, then we have,

For $$ a_0 $$term, suppose$$ 3d_1=a^3$$we have,

For $$b_0$$ term, we have,

So use the trigonometry substitution method, suppose$$ \tan\theta=\tau$$, so we have,

So plug ($$),($$),($$)into ($$), we will have,

So,$$h(x)$$ is given by,

If $$d_1=0$$, then we have,