00:01
Here first we solve part in part we are asked to verify that ux y is a harmonic function or not so let's start to solve this problem we know a harmonic function satisfies laplace equation and laplace equation for a function f is given by del square f equals to 0 and from this we obtained del square f over del x square plus del square f over dahl square f over dela square f over dela square y square equals to 0.
00:34
Now here if f is equal to uxy then del square u over del x square plus del square u over del x square must be equal to zero.
00:53
And here if this expression comes equal to zero then we can say ux y is a harmonic function.
01:01
So here now to find the value of del square u over del x square and del square u over del y square first we find the value of del u over del x so from this we obtain del u over del x is equal to negative 3 to the power negative 3 x into cos 3 y now from this we obtain del square u over del x square is is equal to 9 e to the power negative 3x into cos 3y.
01:41
Now next for this function we calculate del u over del y.
01:47
Now from this we obtain del u over del y is equal to negative 3 e to the power negative 3x sine 3y.
02:04
Now from this we obtained dl square u over dl y square equals to negative 9 to the power negative 3x into sine, sorry, pose 3y.
02:27
Now after adding this and this we obtained dahl square u over delax square plus dl square u over dela y square is equal to 9.
02:40
To the power negative 3x cos 3y minus 9 e to the power negative 3x cos 3y here this and this cancel out so from this we obtained del square u equals to 0 here we see that the u is satisfies la plus equation that's why u is a harmonic function now next we will solve part b in part b, we are asked to find the harmonic conjugate vxy of uxy.
03:20
So let's start to solve this problem.
03:23
Suppose we have fz equals to uxy plus iota vx into y.
03:34
And here we assume that fz is an analytic function.
03:38
And for an analytic function, you and we must satisfy the kochi -riemann equation.
03:46
And according to kochi, ryman equation, we obtain del u over del x is equal to del v over del y and del u over del y is equal to negative del v over del x.
04:04
Now to find v we we have dv is equal to del v over del x d x plus del v over del y d y d y d y.
04:16
Now from the this we obtain dv is equal to now from cr equation we have del v over del x is equal to negative del u over del y d x plus and from cr equation we have del v over del y is equal to del u over del x d y now from this we obtained dv is equal to negative from this we obtain del u over del y is equal to negative e to the power negative 3x, 3, sine 3, dx plus, and from this we obtained del u over del x is equal to negative e to the power negative 3x, cos 3y, d, y.
05:18
Now from this we obtained dv equals to 3 e to the power negative 3x, sine 3y, x minus e to the power negative 3x cos 3y d y.
05:40
Now here this is an exact differential equation because here partial derivative of this with respect to x comes equal to partial derivative of this with respect to y...