Question

Verify `(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)` for u = x³ + 3x² y² + y³.

Answer 1

Given u = x³ + 3x² y² + y³

Differentiating partially with respect to 'y' we get,

`(del "u")/(del "y") = 0 + 3x^2 (2y) + 3y^2 = 6x^2y + 3y^2`

Differentiating again partially with respect to 'x' we get,

`(del^2 "u")/(del x del "y")` = 6y(2x) + 0 = 12xy   ...(1)

Differentiating 'u' partially with respect to 'x' we get,

`(del "u")/(del "x") = 3x<sup>2</sup> + 3y<sup>2</sup> (2x) + 0 = 3x<sup>2</sup> + 6xy<sup>2</sup>`

Differentiating again partially with respect to 'y' we get,

`(del^2 "u")/(del "y" del x)` = 0 + 6x(2y) = 12xy    ....(2)

From (1) and (2)

`(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)`