User:Egm6321.f11.team2.kim/PR4

=R*4.2 - Verify the exactness for L2-ODE-VC =

Given
$$\displaystyle \sqrt{x} \, y'' + 2xy' + 3y = 0 $$  (4.2.1)

Find
Verify the exactness of eq (4.2.1)

Solution
Let $$\displaystyle g(x,y,p) = 2xy' + 3y, \, \, f(x,y,p) = \sqrt{x} $$ $$\displaystyle p = y' $$

$$\displaystyle \underbrace {\sqrt{x}}_{f(x,y,p)}\, y'' + \underbrace {2xy' + 3y}_{g(x,y,p)} = 0 $$ It shows that above equation satisfy the first exactness. In order to satisfy 2nd exacntess condition, $$\displaystyle f_{xx} + 2pf_{xy}+p^2f_{yy} = g_{xp} + pg_{yp} - g_{y} $$ (4.2.2) $$\displaystyle f_xp + pf_yp + 2f_y = g_pp$$ (4.2.3) From the eq (4.2.1),

$$\displaystyle f_{xx} = \frac {1}{4}x^{-\frac{3}{2}}, \, \, f_{xy} = 0, \, \, f_{yy} = 0 $$ $$\displaystyle g_{xp} = 2, \, \, g_{yp} = 0, \, \, g_{y} = 3 \, \, f_{xp} = 0, \, \, f_{yp} = 0, \, \, f_{y} = 0, \, \, g_{pp} = 0 $$ From the above values, eq (4.2.2) and eq (4.2.3) is that $$\displaystyle f_{xx} = g_{xp} - g_{y}$$  (4.2.4)

$$\displaystyle f_{xx} = \frac {1}{4}x^{-\frac{3}{2}}, \, \, \, g_{xp} - g_{y} = -1 $$ $$\displaystyle f_{xp} + pf_{yp} + 2f_{y} = g_{pp} = 0 $$ (4.2.5) This shows that although the eq (4.2.5) satisfies the 2nd exactness condition, eq (4.2.4) does not satisfy the 2nd exactness condition. Thus, the eq (4.2.1) is not exact