R4*1

R*4.1 Checking the Exactness of a L2-ODE-VC

On our honor, we did this problem on our own, without looking at the solutions in previous semesters or other online solutions.

Given

The Given L2-ODE-VC Equation that needs to be tested for its Exactness is Template:NumBlk

Problem

Check (Template:EquationNote) for the Exactness Conditions of a L2-ODE-VC.

Solution

Since the given equation does not have a missing y, the equation has to be for exactness using the Exactness Conditions for a N2-ODE, which are as below:

1st Exactness Condition:

The ODE should be of the form

2nd Exactness Condition Set:

By observing (Template:EquationNote), we find that it satisfies the 1st Exactness Condition as shown below:

Hence, we have identified that,

Let us now test for the 2nd Exactness Conditions. But before that let us find the partial derivative terms $f_{xx},f_{xy},f_{xp},f_y,f_{yy},f_{yp},g_{xp},g_y,g_{yp}$ and $g_{pp}$ required in the conditions individually.

Using (Template:EquationNote), we get,

Using (Template:EquationNote), we get,

Using (Template:EquationNote), we get,

Using (Template:EquationNote), we get,

Using (Template:EquationNote), we get,

Now, plugging (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote) into (Template:EquationNote) which is the 1st equation of the 2nd Exactness Condition Set, we get,

$(\frac{-1}{4}{x}^{\frac{-3}{2}})+(2p)(0)+p^2(0)=2+p(0)-3$

This proves that the given equation is not Exact. However, for a more thorough proof, let us also check the 2nd equation of the 2nd Exactness Condition Set (Template:EquationNote). Plugging (Template:EquationNote), (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote) into (Template:EquationNote), we get,

$0+p(0)+2(0)=0$

Although the 2nd equation of the 2nd Exactness Condition Set holds true, since, the first doesn't, the given ODE cannot be deemed exact. Both the 2nd Exactness Conditions have to hold true for the given ODE to be Exact.

Hence, the given L2-ODE-VC is NOT EXACT.