00:01
In this question, we have to find out the solution of the given differential equation fourth derivative of y plus four times third derivative plus seven times second derivative plus 12 times first derivative plus 12y is equals to zero.
00:18
So, in another form, this can also be written as d4 plus 4d cube plus 7d square plus 12 times of d plus 12 times y is equals to zero where we know that d is equals to just it is representing this operator d over dx.
00:39
So, we can write like this.
00:41
Now, for this, we can write the characteristic equation, which is given as follows m raised to four plus four m cube plus seven m square plus 12 times m plus 12 equals to zero we get.
01:03
Now we have to just simplify this.
01:05
So, observe that m equals to minus two is a solution of this equation is a solution.
01:18
Correct.
01:19
So, what i will do, we can use a synthetic division to find out other factor of this.
01:25
So, let us see how we are going to apply that we can write here the coefficient.
01:31
So, it is one, four, seven, 12 and 12.
01:36
We got one factor as minus two.
01:38
So, i will get here minus two.
01:40
Now minus two times now one minus zero, i will get as one...