00:01
Hi, given y double dash minus 5y dash plus 4y is equal to e rise to the power 2t, also given the condition y of 0 is equal to 2 and y dash of 0 is equal to 11.
00:22
Considering auxiliary equation we have m square minus 5m plus 4 is equal to 0.
00:36
This implies m minus 4 into m minus 1 is equal to 0.
00:44
Implies m is equal to 1 comma 4.
00:49
Therefore the complementary function ycf is given by c1 e rise to the par t plus c2 e rise to the par 4 t.
01:02
Next for particular solution type e rise to the par beta t we have the solution y is equal to a into e rise to the par beta t therefore let consider y p is equal to a into e rise to the power to t differentiating we have yp dash is equal to 2 into a into e rise to the par roti again differentiating we have yp double dash is equal to 2 into 2 4 a into e rise to the par now substituting these values in the equation we have the equation y double dash minus 5y dash plus 4 y is equal to e rise to the power to t substituting the values we have 4a e rise to the par to t minus 5 into 2a he rise to the par to t plus 4 into a e rise to the power 2 t is equal to e rise to the power 2t.
02:13
Taking e rise to the par 2t common, we have e rise to the part 2t into 4a minus 10a plus 4a is equal to e rise to the power to tt.
02:28
This implies we have minus 2a is equal to 1.
02:34
Implies a is equal to minus 1 by 2.
02:38
Therefore we have particular solution yp is given by minus 1 by 2 into e rise to the par 2t.
02:50
Solution y is given by complementary function y cf plus particular solution yp.
02:59
Therefore we have y is equal to c1 e rise to the part t plus c2 e rise to the par 4t minus 1 by 2 e rise to the par 40 minus 1 by 2 e rise to the par 2 t...