00:01
Hi here for the given question we need to find the differential equation, which is y double dash plus a y dash plus b y equals to 0 and here we are given basis as e to the power minus 3 .11 cos of 2 .1 x comma e to the power minus 3 .1 sin of this is 3 .1 x here also 3 .1 x sin of 2 .1 x.
00:31
So here now further we can say that here y of x is equal to a e to the power 3 .1 x multiplied with cos of 2 .1 x plus b multiplied with e to the power minus 3 .1 x multiplied with sin of 2 .1 x.
00:50
So here in our case in order to find the value of above ode we need to take the derivative.
00:57
So here in our case y double dash of x is equal to minus 3 .1 a e to the power minus 3 .1 x cos of 2 .1 x minus 2 .1 a e to the power minus 3 .11 x multiplied with sin of 2 .1 x plus 3 .1 e to the power minus 3 .1 x multiplied with sin of 2 .1 x plus 2 .1 b multiplied with e to the power minus 3 .1 x multiplied with cos of 2 .1 x.
01:32
So here in our case further this is the value of y dash of x.
01:38
Now we need to calculate the value of y double dash of x.
01:41
So here in our case the value of the double derivative will be equal to 6 .48 a minus 4 .41 a multiplied with e to the power minus 3 .1 x multiplied with cos of 2 .1 x plus 6 .48 a plus 4 .41 a multiplied with e to the power minus 3 .1 x multiplied with sin of 2 .1 x plus 9 .6 b minus 4 .41 b multiplied with e to the power minus 3 .1 x multiplied with sin of 2 .1 x plus 9 .6 minus 9 .16 b minus 4 .41 b multiplied with e to the power minus 3 .1 x multiplied with cos of 2 .1 x.
02:43
So here now substituting the value of first derivative of y and second derivative of y in our above equation and simplifying this we can say that here our equation will be equal to 6 .48 a minus 4 .41 a multiplied with plus 6 .6 a minus 4 .41 a multiplied with 1 here also multiplied with 1 plus 9 .61 b minus plus 4 .41 b multiplied with 1 plus 9 .61 b plus 4 .41 this value is negative 4 .41 b and the whole value this whole value is multiplied with e to the power minus 3 .1 x multiplied with sin of 2 .1 x this is equal to 0.
03:48
So here now comparing the coefficient as we know that this value is never going to 0 so we need to compare this coefficient...