00:01
Let's have a look at the question.
00:03
So this is a problem in heat equation for ux t.
00:07
So the given condition is d u by d t is equals to 9 d square u upon d x square.
00:18
Now let us take this as first equation.
00:21
So we have u 0 comma t is equals to du by d x of 2d which is equal to 0.
00:31
And here t is greater than 0.
00:34
And we have u x comma 0 is equals to x square for all x lies between 0 2.
00:44
Now let us take that u x comma t is equal to x x and t t be the solution of equation 1.
00:59
So we have x t dash is equal to 9 x double dash t or we can write this as x double dash upon x is equal to t dash upon 90 let us take this equals to lambda now u zero comma t is equal to zero so we can say that x zero and t t is equals to 0.
01:35
So from here x 0 is equal to 0 and t t does not equals to 0.
01:44
So now we can say du by d x of 2 comma t is equal to 0.
01:52
So that means x dash 2 and t into t is equals to 0.
01:59
That means x dash 2 is also equals to 0.
02:04
Now let us take the case 1 let lambda is equals to 0 therefore x double dash is equals to 0 which means that the solutions will be c1 x plus c2 now as x 0 is equal to 0 that means c2 is equal to 0 so we have x dash 2 is equal to 0 so from here we can say c1 is equal to 0 so we get x x x x x is equal to 0 that means u x t is equal to 0 so this does not satisfy the initial condition so we will reject this case now we will take case 2 here let us assume that lambda is equals to u square greater than 0 so now x double dash is equals to u square x that means x x x x is equals to c1 e to the power ux plus c2 e to the power minus ux now as x 0 is equal to 0 so that means c1 plus c2 is equal to 0 and so from here we can write c2 is equal to minus c1 now also we have x dash 2 is equal to 0 that means u c1 e to the power to u minus use c2, e to the power minus 2u is equals to 0.
03:46
So from here we can write c1 into e to the power 2u plus e to the power minus 2u is equals to 0.
03:55
That means c1 is equals to 0.
03:59
Then c2 will be also equals to 0.
04:03
So that means this case is also rejected.
04:07
Now we will take the third case where lambda is equal...