00:01
The differential equation is dx by dt plus 7x equal to 5 cos 2t.
00:11
So first order linear differential equation is of the form dx by dt plus pt equal to qt.
00:27
So compare with these two we have p of t equal to 7 and qt equal to 5 cos 2t.
00:35
So we will solve this differential equation.
00:38
We will solve this differential equation using integrating factor method.
00:49
So first find the integrating factor that is if say u equal to e raised to power integral of pt into dt which is equal to e raised to power pt is 7 so we have 7 dt and which is equal to integral of 7 dt is 7t.
01:09
Now the solution of the linear differential equation is x dy by dt into x into u equal to qt into u.
01:30
So this implies dy by dt of x into u is e raised to power 70 equal to qt is 5 cos 2t into e raised to power 70.
01:41
Now integrate to both side with respect to t.
01:51
Then we have integration of dy by dt of x into e raised to power 70 equal to integral of 5 e raised to power 70 into cos 2t into dt.
02:02
Since integral and derivative are opposite to each other so we can cancel them.
02:06
So we left with x into e raised to power 70 equal to here 5 is constant so we can take it outside and integral of e raised to power 70 cos 2t into dt.
02:16
Mark this equation first.
02:18
So first we find the integral of e raised to power 70 into cos t.
02:23
So integral of e raised to power 70 cos 2t into dt here we apply the product rule that is the derivative of first function into second function implies the derivative of first function then second remains constant as it is plus the derivative of second function then first remains as it is.
02:40
So using that product rule we have e raised to power 70 into cos 2t equal to e raised to power here we treat e raised to power 7 as second function and this is as our first function.
02:55
So we have e raised to power 70 by into cos 2t by 7 minus integral of minus the derivative of cos 2t is 2 sin 2t into the integral of e raised to power 70 divided by 7...