00:01
In this problem we are provided with the differential equation x double prime of t plus 7 times x of t equals to 8 plus 2 times cos of 2 pi t plus 9 times cause of 3 pi t and we are provided with the initial conditions x prime of 0 equals to 0.
00:32
And x prime of 1 equals to 0.
00:36
We are asked to find out the solution for this problem.
00:41
That is, we need to find out x of t.
00:44
So first let us begin by writing down the characteristic equation.
00:47
So we have lambda squared plus 7 equals to 0, which implies that lambda equals to plus or minus square root of 7 times iota.
00:57
So in this case, the general solution x bar is given by c1, times cost of square root of 70 plus c2 times sine of square root of 70.
01:11
Let us mark this as equation number one.
01:14
Next, we begin by finding out the particular solution, starting with the particular solution for 9 times cost of 3 pi t.
01:26
So by making use of the method of undetermined coefficients, we have, sorry, x, xp1 to be of the form a times cause of 3 pi t plus b times sign of 3 pi t so substituting this in the provided differential equation we obtain the value of a as negative 9 over 9 pi squared minus 7 and we obtain the value of b to be 0 so substituting this we have xp 1 to be equal to negative negative 9 cos of 3 pi t the whole divided by 9 pi squared minus 7.
02:12
So let us mark this as equation number 2.
02:15
Next we find out the particular solution for 2 times cause of 2 pi t.
02:25
So in this case the particular solution is of the form a times cause of 2 pi t plus b times sine of 2 pi t.
02:40
So substituting this in the given differential equation, we obtain the value of a to be negative 2 over 4 pi squared minus 7 and we obtain the value of b to be 0.
02:55
So substituting this we have xp2 to be equal to negative 2 cost of 2 pi t the whole divided by 4 pi squared minus 7...