User:Egm6321.f12.team2.mohan/Report4

Verify that
is an exact L2-ODE-VC.

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

First Exactness Condition
The first exactness condition is as follows, where $$p=y'$$

To verify for exactness, ($$) can be written as follows, so ($$) meets the first exactness condition for N2-ODE's.

Second Exactness Condition
The second exactness condition is as follows, and From the above two equations we can see that, {| style="width:100%" border="0" $$f_{xx} = \frac {\partial (\sqrt x)}{\partial x \partial x} = -\frac{1}{4 x^{3/2}}$$ $$f_{xy} = \frac {\partial (\sqrt x)}{\partial x \partial y} = 0$$ $$f_{yy} = \frac {\partial (\sqrt x)}{\partial y \partial y} = 0$$ $$f_{xp} = \frac {\partial (\sqrt x)}{\partial x \partial p} = 0$$ $$f_{yp} = \frac {\partial (\sqrt x)}{\partial y \partial p} = 0$$ $$f_{y} = \frac {\partial (\sqrt x)}{\partial y} = 0$$ $$g_{xp} = \frac {\partial (2xp+3y)}{\partial x \partial p} = 2$$ $$g_{yp} = \frac {\partial (2xp+3y)}{\partial y \partial p} = 0$$ $$g_{y} = \frac {\partial (2xp+3y)}{\partial y} = 3$$ $$g_{pp} = \frac {\partial (2xp+3y)}{\partial p \partial p} = 0$$
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so, plugging the above values into equations ($$) and($$) respectively, we get, Although the L2-ODE-VC meets the second portion of the second exactness condition, it does not meet the first portion, so the equation is not exact.