Question

Clairaut's theorem holds that Uxy=Uyx,show that the following equations obey clairaut's theorem.

1.u=in(x+2y).

2.u=e^xy siny.

Answer 1

1)$U = ln(x+2y)$

$U_x = \frac{1}{x+2y}\\ U_{xy} = \frac{-2}{(x+2y)^2}$

Also,

$U_y = \frac{2}{x+2y}\\ U_{yx} = \frac{-2}{(x+2y)^2}$

Clairaut's theorem as being justified as desired

2) $U = e^{xy} \sin y$

$U_x = ye^{xy}\sin y \\ U_{xy} = e^{xy}\sin y + + xy e^{xy}\sin y + ye^{xy}\cos y$

$U_y = xe^{xy}\sin y + e^{xy}\cos y\\ U_{yx} = e^{xy}\sin y + + xy e^{xy}\sin y + ye^{xy}\cos y$

Clairaut's theorem as being justified as desired.