00:01
This equation we have y double dash minus 2y dash plus y is equals to 6 e raised to the power x divided by x cube.
00:10
We have to solve the given second order differential equation.
00:17
So, you have to solve the given differential equation.
00:22
Now, let us focus on our solution.
00:25
So, we know that this is a non -homogeneous equation given to us.
00:29
So, it has two parts in its solution.
00:32
It has a homogeneous part homogeneous solution and it has a particular solution.
00:42
So, we can write particular solution as yp and homogeneous solution as yh.
00:47
So, the general solution y of x is equals to y homogeneous part plus the particular solution.
00:55
So, for homogeneous let us first solve the homogeneous part.
01:00
So, we have the homogeneous differential equation as y double dash minus 2y dash plus y is equals to 0.
01:07
So, this implies you have lambda square minus 2 lambda plus 1 is equals to 0.
01:12
This will be the characteristic equation.
01:15
So, this implies you have lambda minus 1 is equals to 0.
01:21
So, we have lambda is equals to 1 as a repeated root.
01:26
So, this is a repeated root for the given characteristic equation.
01:31
So, we have y1x and y2x.
01:34
These are the two different solutions for the homogeneous part.
01:39
So, we have e raised to the power x as the first solution and the second solution will be x times y1x.
01:46
So, this solution has been found out using the reduction of order method...