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Question
For the differential equation given, find a particular solution satisfying the given condition:
`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`
Solution
The given equation is
`dy/dx + 2y tan x = sin x`
Which is a linear equation of the type
`dy/dx + Py = Q`
Hence P = 2 tan x and Q = sin x
∴ `int Pdx = int 2 tan x dx = 2 log |sec x| = log sec^2 x`
∴ `I.F. = e^(int Pdx) = e^(log sec^2x) = sec^2 x`
∴ The solution is `y. (I.F.) = int Q. (I.F.) dx + C`
⇒ `y sec^2 x = int sin x sec^2 x dx + C`
`= int sec x tan x dx + C`
⇒ `y sec^2x = sec x + C`
When `x = pi/3, y = 0; "then" 0 = sec pi/3 + C`
⇒ C = -2
Putting in (1), y sec2 x = sec x - 2
⇒ y = cos x - 2 cos2x,
Which is the required solution.
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