Question

If u = e^xy, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x² + y²).

Answer 1

Given, u = e^xy

Differentiating partially with respect to x, we get,

`(del"u")/(del"x")` = y e^xy (Treating y as constant)

`(del^2"u")/(delx^2) = del/(delx) ("y"e^(xy))`

`= "y" del/(delx) (e^(xy))`

= y(ye^xy)

= y²e^xy ……… (1)

We have u = e^xy

Differentiating partially with respect to y,

`(del"u")/(del"y")`= x e^xy

Again differentiating partially with respect to x, we get,

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

`= "x" del/(delx) (e^(xy))`

= x²e^xy ……… (2)

Adding (1) and (2) we get,

`(del^2"u")/(delx^2) + (del^2"u")/(dely^2)` = e^xy(x² + y²)

= u(x + y ) [∵ u = e^xy]