00:01
Hello everyone, in this problem we are given with the differential equation d square y by dt square minus 10 dy by dt plus 9y to be equal to 5t with the condition y of 0 to be equal to minus 1 and dy of dt of 0 to be equal to 2.
00:24
So, that is y dash of 0 equal to 2.
00:26
Now we need to solve this equation by using laplace.
00:30
We are going to solve this differential equation.
00:32
Now taking laplace on both sides of the given differential equation we get l of d square y by dt square minus 10 dy by dt plus 9y which is equal to l of 5t.
00:53
So, now we can rewrite this to be l of d square y by dt square minus 10 l of dy by dt plus 9 l of y to be equal to 5 multiplied by l of t.
01:14
So now taking laplace by using laplace formula we can rewrite this to be s square of 5 minus s y of 0 or here it is s square of y of s minus y of 0 minus minus y of 0 minus 10 of this can be written as s multiplied by y of s minus y of 0 plus 9 multiplied by l of y can be written as y of s which is equal to 5 multiplied by l of t can be written as 1 by s square.
02:01
So, now simplifying this we can write this to be s square minus 10 s plus 9 multiplied by y of s plus 5 minus 12 to be equal to 5 by s square.
02:16
So, from this we get the value of y of s to be equal to 5 divided by s square multiplied by s square minus 10 s plus 9 minus of s sorry this is not s sorry this is not 5 this is s s minus 12 divided by s square minus 10 s plus 9.
02:45
Now let us consider this value to be y1 and this value to be y2.
02:49
So, now consider y1 which is 5 divided by s square multiplied by s square minus 10 s plus 9...