Tutor profile: Bruce C.

What is the partial derivative with respect to x of the following expression, $(yxcos(ax)$)

Since this is a partial derivative we treat all variables other than x as a constant. We then have two parts with x in them, and apply the product rule to each component. $(ycos(ax) - yxasin(ax)$)

If there is no circulation in a fluid flow, what is the vorticity at any point in the flow?

By the definition of circulation, $$\Gamma$$, the vorticity is zero at all points in the flow.

Find the general solution, $$y(x)$$.$( y'' - 4y' +4y' = 0$)

This is a second order linear homogeneous differential equation, so we can solve easily by finding the roots of the auxiliary equation. This has roots in the form, $(m^2-4m+4=(m-2)(m-2) $) Because there are identical repeat roots, the complimentary solution is in the form, $( y_c=c_1e^{2x} + c_2xe^{2x}$)