Question

An equation relating to the stability of an aeroplane is given by ` (dv)/(dt) = g cos alpha - kv`, where ` v` is the velocity and ` g, alpha, k` are constants. Find an expression for the velocity if `v = 0` at ` t =0`

Answer 1

The given differential equation is
` (dv)/(dt) + kv= g cos alpha " "` ... (i)
This is of the form ` (dv)/(dt) + Pv = Q`, where `P = k and Q = g cos alpha `.
Thus, the given equation is linear.
`IF = e ^(int Pdx) = e ^(int k dt ) = e^(kt )`
So, the solution of the given differential equation is
`vxx IF = int {Q xx IF}dt + C`,
i.e., `ve^(kt) = int (gcos alpha ) e^(kt) dt + C`
`" " = ((g cos alpha ) e^(kt))/( k ) +C" " `... (ii)
Now, it is given that ` v =0` when `t = 0`
Putting `t=0 and v=0 `in (ii), we get `C = (-g cos alpha)/( k)`
` therefore ve^(kt) = ((g cos alpha ) e^(kt))/(k) - (gcos alpha )/( k)`
`rArr v = (1)/(k) (g cos alpha )( 1 - e^(kt))`, which is the required expression.
SOLUTION OF `(dx)/(dy) + Px =Q`
Working Rule for Solving ` (dx)/(dy) +Px = Q `
(i) Find If = ` e ^(int Pdy)`
(ii) The solution is given by ` x xx IF = int {Q xx IF}dy +C`.