00:01
Hi, in this question we need to solve the given initial value problem.
00:05
Y power 3 plus 3y double dash minus 10y dash equals 0.
00:11
Also given initial condition is y of 0 equals 7, y dash of 0 equals 0 and y double dash of 0 equals 70.
00:20
Here first assume that y of x equals e power lambda x so that d cube divided by dx cube into e power lambda x plus 3 d square divided by dx square e power lambda x minus 10 into d by dx e power lambda x equals 0.
00:50
On differentiating then we get lambda cube e power lambda x plus 3 lambda square e power lambda x minus 10 lambda e power lambda x equals 0.
01:04
On taking e power lambda x commonly outside then we get e power lambda x into lambda cube plus 3 lambda square minus 10 lambda equals 0.
01:14
Here e power lambda x must not be equal to 0.
01:20
Next we need to find the roots so that lambda cube plus 3 lambda square minus 10 lambda equals 0.
01:32
On solving this we get lambda into lambda minus 2 into lambda plus 5 equals 0.
01:39
Therefore lambda equals minus 5 lambda equals 0 or lambda equals 2.
01:45
Therefore general solution is y of x equals y1 of x plus y2 of x plus y3 of x.
02:07
Here y1 of x equals c1 e power minus 5 x and y2 of x equals c2 e power 0 and y3 of x equals c3 e power 2 x.
02:35
Therefore we can write it as the general solution y of x equals c1 e power minus 5 x plus c2 plus c3 e power 2 x...