User:Egm6321.f12.team4.hong/Report4

Problem R*4.1 – Verifying Exactness of L2-ODE-VC
sec21-1

Statement
Verify the exactness of the following equation,

Solution
To solve this problem, we have to know the two exactness conditions for second order ODEs. The equation must be in the following form
 * First Exactness Condition for N2-ODEs:


 * Second Exactness Condition for N2-ODEs:

Let's rearrange the equation 4.1.0

If we let, $$ g(x,y,p) = 2xy' + 3y $$ and $$ f(x,y,p) = \sqrt(x) $$ then we can have our equation in the form exactness condition one as stated above. Therefore, the exactness condition one is satisfied.

To apply second exactness condition we need to find $$ f_{xx}, f_{xy}, f_{xp}, f_y, f_{yy}, f_{yp} $$ and $$ g_y, g_{xp}, g_{yp}, g_{pp} $$ They are found easily by partial differentiation and the results are as follows,

Now if we apply the first and second equation of the Second Exactness Condition

We can see that the first equation of the second exactness condition does not satisfy where as the second equation does satisfy. Unfortunately, both equations must be satisfied to satisfy the second exactness condition. Therefore, the second exactness condition is not satisfied such that the equation 4.1.0 is not exact.

Author and References

 * Solved and Typed by -- Seong Hyeon Hong
 * Reviewed by --