00:01
Hi, in this problem we have y double dash minus 2y dash plus 5y equal to 0.
00:10
Now we need to find the linear independent solution y1 and y2 for this differential equation.
00:19
Also we need to find the ron skin, the general solution and the particular solution.
00:24
Now the auxiliary equation for the differential equation is going to find the wrong skin.
00:31
Given by m square minus 2m plus 5 equal to 0 and this further implies that m is equal to 1 plus minus 2 aota so therefore with the help of this the complementary function that is cf for the differential equation is equal to e power x into c1 cos 2x plus c2x plus c2 2, sine 2x.
01:07
Therefore, the 2 linearly independent solution, that is, y1 is equal to e power x, cos 2x, and y2 2 equal to e power x, sine 2x.
01:24
So now further, the ronskin is given by the formula, w equal to determinant of y1, y2, y1 dash y2 dash.
01:39
Therefore this is equal to determinant of e power x cos 2x, e power x sine 2x and e power x into cos 2x minus 2 sine 2x and e power x sine 2x plus 2 cost 2x.
01:56
Therefore simplifying the expression we obtain w equal to 2 e power 2x 2x 2 e power 2x and further, this is not equal to 0...