User:EGM6321.f12.team7.Marjanovic/report3/3.4

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Problem 4: Finding the Integrating Factor h to make a L1-ODE-VC exact
Based on lecture notes section 12

Given: Exactness Conditions and the Integrating Factor
A General Nonlinear 1st Order Ordinary Differential Equation is exact if it satisfies the following two exactness conditions (from section 8 and section 9) : 1st Condition: An N1-ODE has to take the form:

2nd Condition: The partial derivatives of the N1-ODE:

If the N1-ODE satisfies the 1st condition ($$) but not the 2nd condition ($$), there exists an integrating factor $$\displaystyle h(x,y) $$ such that the following N1-ODE is exact (from section 10) :

Furthermore $$\displaystyle h(x,y) $$ is only a function of x if the following 3 conditions are true (from section 12) :

Given all of those conditions, the integrating factor can be written as:

Find: Integrating Factor h to make the L1-ODE-VC exact
For the following L1-ODE-VC, if it is not exact, find the integrating factor h to make it exact:

Solution: Integrating Factor h(x)

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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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Though this is a L1-ODE-VC, we can use the Integrating Factor Method for general N1-ODEs to make it exact. To begin, we must first test the two exactness conditions ($$) and ($$)

For the first condition, let

From here we can see that

For the second condition,

We can see that

Therefore ($$) is satisfied but ($$) is not.

Furthermore, we can see that ($$), ($$), and ($$) are all functions of x, thus the integrating factor is going to be:

Thus the integrating factor is

Where k is the integration constant.