00:01
We are given a double differential equation y double dash plus 4y is equals to 2 cos 2x and we are given that y of 0 is equals to 0 and y dash pi by 2 is equals to 0.
00:17
We can see that this is a non -homogeneous differential equation of second order.
00:23
So let general solution be general solution of this equation be y of x.
00:34
We have y of x is equals to solution of the homogenous equation plus the particular solution of the given differential equation.
00:45
So let us first find the homogenous equation.
00:48
Solution to the homogenous equation.
00:50
The homogenous equation will be y double dash plus 4y is equals to 0.
00:55
This is the homogenous equation of the given differential equation.
00:59
The characteristic equation will be lambda square plus 4 is equals to 0 and we have lambda is equals to plus minus 2 iota.
01:07
So this is the value for lambda.
01:10
So therefore we have the homogenous solution as a cos 2x plus b sin 2x.
01:20
This will be the homogenous solution and the basis will be y1 is equals to cos 2x and y2 is equals to sin 2x.
01:34
These two solutions form the basis for the homogenous equation.
01:40
Now let us find the particular solution.
01:42
So formula for the particular solution is minus y1 integral y2 multiplied by r upon w dx plus y2 integral y1 multiplied by r upon w dx.
01:59
The equation we have r is equals to 2 cos 2x.
02:04
Now let us find the wronskian.
02:06
Wronskian will be the determinant value of cos 2x and here we will write sin 2x...