User:Cloudwu890202/Report 3.7

=R*3.7=

Problem 7:Find the exactness and Find the first integral
Report problem 3.7 from lecture notes section 13-4.

Given: A specified function
Composites of specific function is shown as below

Find: show the exactness and find the first integral
Set up a function in terms of given composites, and make it exact, or show that it could be made exact by the IFM, then find the first integral

Solution: examine the function with 1st and 2nd condition and find the first integral

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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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Calculate other composites of this function

Set up a function in terms of given composites as below

when

Thus, function can be written as

To examine whether the function is exact is to find out that whether the function is satisfied 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

Then find the n(x) as below, assuming h is a function of x only

h(x) can be calculate as below

So the Eqn. 7.10 can be changed into

with

This function is exact by IFM

By definition

with

and

Calculation shown as below

Thus