User:EGM6321.f12.team4.Rui.C/Homework3 R3.12

Statement
1. Derive (2)p.16-5, the 2nd exactness condition, by differentiating the definition of $$g(x,y,y') $$in (3)p.16-4 with respect to $$p:=y'$$ defined in (2)p.7-3

2. Derive (1)p.16-5, the 1st relation in the 2nd exactness condition.

3. Verify that (1)p.16-6 satisfies the 2nd exactness condition.

Solution for (1)
The 2nd exactness condition (2)p.16-5 can be written as:

Since

where

(3.12.2) can also be written as:

where

Take the derivative of $$g(x,y,p)$$ with respect to $$p$$:

Since

and substitute (3.12.6) into (3.12.8) and (3.12.8), we have:

Substitute (3.12.10) and (3.12.11) into (3.12.7):

Take the second derivative of $$g(x,y,p)$$ with respect to $$p$$:

Substitute (3.12.11) into the above equation, we have

Finally, we have

which is the required equation to prove.

Solution for (2)
The 1st exactness condition (1)p.16-5 can be written as:

From (3.12.5):

Take the derivative of both sides of (3.12.16):

Substitute (3.12.10) and (3.12.11) into the above equation:

Then we have:

From (3.12.5) and (3.12.11):

Substitute (3.12.21) into (3.12.22) for $$\phi_y$$:

which can be reorganized as:

Using

and substitute (3.12.24),(3.12.20) into the above equation, we have:

After reorganization, we have

which is the required 1st exactness condition.

Solution for (3)
(1)p.16-6 is:

Then we can find that

For the first exactness condition:

in which we have

Substitute (3.12.35) into (3.12.36), we have:

So the first exactness condition is satisfied.

For the second exactness condition:

Substitute (3.12.35) and (3.12.38) into (3.12.37):

So the second exactness condition also holds. Proved.

Author and References

 * Solved and Typed by -- Rui Che