00:01
This question has given you various functions u of x and y.
00:04
It has to determine whether each of them is a solution of laplace's equation which is uxx plus uyy equals 0.
00:14
Now for the first one we've got u equals x squared plus y squared.
00:21
So if we work out uxx is equal to 2 and uyy is also equal to 2.
00:37
So uxx plus uyy is equal to 4 which is not 0.
00:45
So this first one isn't a solution of laplace's equation.
00:51
Now for the second one we've got u equals x squared minus y squared.
00:56
So if we work out uxx is equal to 2 again.
01:01
This time we've got uyy equal to minus 2.
01:06
So this time if we add them together uxx plus uyy we do get 0.
01:14
So this second one is a solution of laplace's equation.
01:23
Now for the third one u equals x cubed plus 3xy squared.
01:32
If we work out uxx the first part will differentiate to 6x and the second part will differentiate to 0.
01:45
If we work out uyy the first part will differentiate to 0 and the second part will differentiate to 6x.
01:54
So uxx plus uyy is equal to 12x here and that's not equal to 0.
02:09
So this third one isn't a solution of laplace's equation...