"Multiplying" the entire equation by dx gives
An equation of the form M(x,y)dx+ N(x,y)dy= 0 is "exact" if and only if
Here so [tex]M_y= 18xy^2- sin(y)+ 6kx^2y^2- xcos(y)[tex] while so . There is NO value of k that will make that exact.
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In that case so and so . Those will be equal and the equation will be exact if 4k= 18 or if k= 9/2.
In that case, there exist F(x,y) such that . We must have . Integrating that with respect to x while holding y constant, . The "constant of integration", since we are holding y constant, may be some function of y, p(y).
Now, differentiating that with respect to y, and that must be equal to which, since k= 9/2 says so that everything involving x cancels (as it must) and we have p'(y)= y. From that where C now really is a constant.
The original equation says that dF= 0 which means F is equal to a constant: and, combining the fractions, .