00:01
Hello, let's have a look at the question.
00:02
So the question state that the differential equation below represent a forced spring mass system without damping in which the mass is m which is equals to 4 kg, the spring constant k is equals to 100 kg per second square and the forcing function is ft which is equals to 1000 cos 5t into motion by pulling the spring down 3 .5 meter.
00:39
So 3 .5 meter is the spring pull down and releasing it.
00:45
So we have the differential equation as 4y double dash plus 100y is equals to 1000 cos 5t and we have that y is 0 is equals to 3 .5 and y dash at 0 will be equals to 0.
01:05
So here for the a part we need to find the general solution to the homogenous equation that is the complementary solution and we will use c1 and c2 for the unknown constant c1 and c2.
01:21
So here let us take that y is equals to e to the power mt be the trial solution of the homogenous equation.
01:39
So therefore we can write that 4m square plus 100 is equals to 0 divided by 4 we will get m square plus 25 is equals to 0.
01:51
So m will have the value plus minus 5 ita.
01:56
Now we can say that the complementary function can be written as c1 cos 5t plus c2 sin 5t.
02:09
Now for the b part we have to find any particular solution to the non -homogenous equation.
02:18
So for ft is equals to here we can divide by 4 so we get 250 in place of 1000 and here we have cos 5t.
02:32
Now ypt is the one of the form a cos 5t and plus b sin 5t but this is also the solution of homogenous equation.
02:48
Thus we can say that ypt is equals to t multiplied with a cos 5t plus b multiplied with sin 5t.
03:02
Now we can say that the differentiation of ypt will be equals to a cos 5t plus b sin 5t and then we will have plus t multiplied with minus 5a sin 5t plus 5b cos 5t.
03:32
Also we can find the value for y double dash pt which will be equals to 5 multiplied with minus 5a sin 5t and plus 5b cos 5t then we will have plus t multiplied with minus 25a cos 5t and minus 25b sin 5t.
04:04
So now we will have that y double dash pt plus 25y is equals to 250 cos 5t.
04:19
So therefore we will put the values in this equation so we will get minus 25a sin 5t plus 25b cos 5t then minus 25t a cos 5t minus 25t b sin 5t then we will have plus 25t a cos 5t and plus 25t b sin 5t and this is whole equals to 250 cos 5t.
05:03
So here on simplifying we will get minus 25a sin 5t plus 25b cos 5t and this is equals to 250 cos 5t...