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प्रश्न
If u = exy, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x2 + y2).
उत्तर
Given, u = exy
Differentiating partially with respect to x, we get,
`(del"u")/(del"x")` = y exy (Treating y as constant)
`(del^2"u")/(delx^2) = del/(delx) ("y"e^(xy))`
`= "y" del/(delx) (e^(xy))`
= y(yexy)
= y2exy ……… (1)
We have u = exy
Differentiating partially with respect to y,
`(del"u")/(del"y")`= x exy
Again differentiating partially with respect to x, we get,
`(del^2"u")/(dely^2) = del/(del"y")`(x exy)
`= "x" del/(delx) (e^(xy))`
= x2exy ……… (2)
Adding (1) and (2) we get,
`(del^2"u")/(delx^2) + (del^2"u")/(dely^2)` = exy(x2 + y2)
= u(x + y ) [∵ u = exy]
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