00:01
Hello students, to find the general solution to the given second -order linear homogeneous differential equation y' ' plus 4y ' plus 4y equal to t to the power minus 2 into e to the power minus 2t, we first need to find the compulsory function for the homogeneous part of the equation and then find the particular integral for the non -homogeneous part of the equation.
00:25
So for the complementary function y' ' plus 4y ' plus 4y equal to 0.
00:33
So putting y equal to e to the power rt, we have r square e to the power rt plus 4r e to the power rt plus 4e to the power rt equal to 0.
00:47
So e to the power rt, r square plus 4r plus 4 equal to 0.
00:54
So r square e to the power rt equal to 0 can be 0.
00:58
So r square plus 4r plus 4r plus 4 equal to 0.
01:06
So r plus 2 whole square equal to 0.
01:10
So r equal to minus 2, r equal to minus 2 and minus 2 double root.
01:17
So the complementary function equal to c1 e to the power minus 2t plus c2t e to the minus 2t.
01:25
Now need to find the particular integral.
01:28
So guessing a particular integral that is yp equal to a t to the power minus 2 e to the power minus 2t...