User:EGM6321.f12.team4.Rui.C/Homework3 R3.6

Statement
Verify that (1)13-4

either is exact, or can be made exact by the IFM and find the integrating factor h.

Solution
Let

Then the N1-ODE can be written in the particular form:

So it satisfies the first exactness condition.

For the second exactness condition:

It is clear that

So the second exactness condition does not hold without the IFM.

Assume an IF $$h(x,y)$$ to let the second exactness condition hold:

Take the partial derivative with either x or y on each side of the equation:

Assume that $$h(x,y)=h(x)$$. The expressions for the above partial derivatives are simplified to:

Let the above two expressions equal. We have:

where

So

Thus, the N1-ODE can be made exact by the IFM and the h(x) is derived.

Author and References

 * Solved and Typed by -- Rui Che