Question

can somebody help me with EXACT DIFFERENTIAL

Answer 1

\[(ye^{xy} + 3x^{2})dx + (xe^{xy} - \cos y)dy \]

Answer 2

i think your job is to check that \[\left(\frac{\delta A}{\delta y}\right)_x=\left(\frac{\delta B}{\delta x}\right)_y\]

Answer 3

i actually have my answer but i doubt about it. just checking.

Answer 4

that is, take the partial of \(ye^{xy}+3x^2\) wrt to \(y\), then wrt \(x\)
similarly take the partial of \(e^{xy}-\cos(y)\) wrt \(x\) then wrt \(y\) and see if they are equal

Answer 5

\[xye^{xy}+e^x{xy}\] for the first one wrt to \(y\) the wrt \(x\) \(xy^2e^{xy}+2e^{xy}\) if i did it right

Answer 6

Example of a similar problem:
\[(-ye^{xy}-\sin{x}+12x^{2}y^{2})dx+(14yx^{3}-xe^{xy})dy\]This is the form\[df=Mdx+Ndy\]First is the need to check for exactness, the equation is exact if and only if\[\frac{dM}{dy}=\frac{dN}{dx}\]In this case\[\frac{dM}{dy}=\frac{dN}{dx}=(-xy-1)e^{xy}+42x^{2}y\]so the equation is exact.

Then we find\[\int\limits_{}^{}Mdx\]and\[\int\limits_{}^{}Ndy\]\[\int\limits_{}^{}Mdx=\int\limits_{}^{}-ye^{xy}-\sin{x}+21x^{2}y^{2}dx=-e^{xy}+\cos{x}+7x^{3}y^{2}\]and\[\int\limits_{}^{}Ndy=\int14x^{3}y-e^{xy}dy=7x^{3}y^{2}-e^{xy}\]Then we "combine" them: each term from each answer is written once and only once in our final answer.

So we would have\[7x^{3}y^{2}-e^{xy}+\cos{x}\]since the 7x^3y^2 and -e^xy terms appear in both, but the cos(x) only appears in one.

Then we just add the constant of integration c to get the final answer:\[7x^{3}y^{2}-e^{xy}+\cos{x}=c\]