00:01
Hello everyone.
00:03
We are given an e t equation which is represented by do you over do t is equal to k times do square u over do x square.
00:12
And the initial condition given to us are u of x comma 0 is equal to 6 times sine pi x divided by l, u of 0 is equal to 0 and u of l comma t is equal to 0.
00:25
We need to find the solution of the e .t equation.
00:29
So let us represent the solution to be u of x comma t is equal to x of x and t of t.
00:41
So upon differentiating with respect to t, we will get ut is equal to k times of uxx.
00:52
This is nothing but the given equation.
00:54
So let us differentiate this.
00:56
We will get x of x t dash of t is equal to k times of x double dash of x t of t of t so next step what we are going to do is we are going to divide this by x of x and t of t that is so dividing it over x of x and t of t on both sides so therefore this would imply that t dash of t divided by t of t is equal to k times x double dash of x divided by x of x.
01:44
And this we are going since both are equal, they must be equal to some constant.
01:49
Let us represent that by lambda.
01:54
So let us, after doing this, what we are going to do is we are going to use the initial so the first initial condition is u of x comma 0 is equal to 6 sine pi x divided by l.
02:13
So proceeding further, we see that when we use these initial condition we get some sort of equations...
