Question

For the differential equation given, find a particular solution satisfying the given condition:

`dy/dx - 3ycotx = sin 2x; y = 2`  when `x = pi/2`

Answer 1

The given equation is 

`dy/dx - 3 y cot x = sin 2x`                ....(1)

Which is a linear equation of the type

`dy/dx + Py = Q`

Here P = - 3cot x and Q =  sin 2x

∴ `intP dx = -3 int cot x  dx = -3 log |sin x|`

∴ `I.F. = e^(-3log|sin x|)`

`= e^(log cosec^3 x)`

`= cosec^3 x`

∴ The solution is `y. (I.F.) = int Q. (I.F.)  dx + C`

`y cosec^3 x = int sin2x cosec^3x dx + C`

`= int (2 sin x cos x)/(sin^3 x)  dx + C`

`= 2 int cosec x cot x  dx + C`

`= - 2 cosec  x  +C`

⇒ y = -2 sin² x + C sin³ x                          ....(2)

When `x = pi/2, y = 2`

∴ `2 = -2 sin^2  pi/2 + C sin^3  pi/2`

⇒ 2 = -2 (1)² + C (1)³

⇒ C = 2 + 2

⇒ C = 4

Putting in (2), we get

y = - 2sin² x + 4 sin³ x

⇒ y = 4 sin³ x - 2 sin x

Which is the required solution.