00:01
Hi there, in this problem the initial value problem is given and we have to solve for y of x.
00:06
So the differential equation is y double dash minus 8 by dash plus 16y is equal to 8 multiplied by e -raise to the power 2 t.
00:17
So we will firstly find the complementary solution of this differential equation, complementary solution which can be determined by solving y double dash minus 8y dash plus 16y is equal to 0.
00:43
So this can be written as d squared minus 8d plus 16 multiplied by y is equal to 0.
00:57
This is of the form fd is equal to 0 where fd is d squared minus 8d plus 16.
01:12
So the characteristic equation, the characteristic equation is given by f of r is equal to 0.
01:38
So we have r squared minus 8r plus 16 is equal to 0.
01:44
This can be written as r minus 4 whole whole square is equal to 0.
01:50
And from here we get the values of r.
01:53
R is equal to 4 comma 4.
01:55
So the complementary solution is given by yt is equal to c1 e raise to the power r that is 4t plus c2 t e raised to the power 4t this can be differentiated and the derivative is given as y -dh t is equal to 4 multiplied by c1 e raised to the power 4 t plus c2 multiplied by c2 e raised to the power 40 plus 4 multiplied by c2 t e raised to the power 40.
02:44
This is equation 1 and equation 2.
02:48
It is given in the problem that y of 0 is equal to 3.
02:58
So this gives equation 1 as y of 0 is equal to c1 multiplied by e raised to the power 4 multiplied by 0.
03:15
Plus c2 multiplied by 0 multiplied by e raised to the power 4 multiplied by 0...