00:01
Hi, here for the given question, we are given that here.
00:05
We have differential equation is y double dash plus 4 y is equal to 8 t square minus 20 t plus 8.
00:13
So here in our case now, if we consider the homogeneous part, it will be equal to y double dash plus 4 y equals to 0.
00:22
Now rewriting this in terms of auxiliary equation.
00:24
It can be written as lambda square plus 4 is equal to 0.
00:28
So here now on simplifying this we have eigenvalue.
00:31
Lambda equals to plus or minus 2 iota.
00:34
So here the complementary solution or the homogeneous solution can be written as c1 sin 2x plus c2 sin 2x.
00:45
So here this is our first solution and here it will be cos 2x and here it is sin 2x.
00:51
Let this be our equation number one.
00:53
Now we need to find the value of the particular solution.
00:56
So here in order to calculate the value of the particular solution.
00:59
Now, let us consider equation 1 and here we know that let from equation 1 y1 is equal to cos 2x and y2 is equal to sin 2x.
01:12
Now here differentiating this we have y1 dash is equal to minus 2 sin 2x and y2 dash is equal to 2 cos 2x and here f of x is equal to 8 t square minus 20 t plus 8.
01:29
So here now further using this two equation, we know that here we have value as c dash of x multiplied with cos 2x plus c1 dash of x multiplied with sin 2x is equal to 0 and minus 2 c dash of x multiplied with sin 2x plus 2 c1 dash of x multiplied with cos 2x is equal to 8 t square minus 20 t plus 8.
02:04
So here now further using this value, we need to use the method of undetermined coefficient.
02:10
So we will find the bronskian of the function.
02:12
So w is equal to cos 2x sin 2x minus 2 sin 2x 2 cos 2x.
02:24
So here now solving this determinant then bronskian equals to 2 sin square 2x plus 2 cos square 2x.
02:36
Now further here in our case, we know that in the similar manner we can find the value of bronskian 1 which is equal to 0.
02:46
Here we have sin 2x again 8 t square minus 20 t plus 8.
02:53
Here we have 2 cos 2x and here calculating this determinant.
02:58
We have value equals to minus 8 t square sin 2x plus 20 t cos 2x.
03:09
Here the value will be sin 2x minus 8 sin 2x.
03:22
Now similarly bronskian 2 can be calculated as cos 2x 0 minus 2 sin 2x 8 t square minus 20 t plus 8.
03:34
So calculating this value we have value equals to 8 t square cos 2x minus 20 t cos 2x plus 8 cos 2x.
03:46
So here now these are the values now here using this values further we can find the value of the constant.
03:53
So here c dash of x is equal to w1 upon w...