Question

Solution of the differential equation `dy/dx = ((tanx - y))/(cos^2x)` is ______ 

पर्याय

• `y = tanx - 1 + ce^{-tanx}`

• `y^2 = tanx - 1 + ce^{tanx}`

• `ye^{tanx} = tanx - 1 + c`

• `ye^{-tanx} = tanx - 1 + c`

Answer 1

Solution of the differential equation `dy/dx = ((tanx - y))/(cos^2x)` is `underline(y = tanx - 1 + ce^{-tanx})`.

Explanation:

`dy/dx = ((tanx - y))/(cos^2x)`

⇒ `dy/dx = tanx sec^2x - ysec^2x`

⇒ `dy/dx + ysec^2x = tanx sec^2x`

Here, P = `sec^2x`, Q = `tanx sec^2x`

∴ I.F. = `e^{int sec^2x dx} = e^{tanx}`

∴ solution of the given equation is

`y . e^{tanx} = int tanx . sec^2x e^{tanx} dx + c`

Put tan x = t ⇒ sec²x dx = dt

∴ `y e^{tanx} = int t e^t dt + c`

⇒ `y e^{tanx} = t e^t - e^t + c`

⇒ `y e^tanx = e^tanx(tanx - 1) + c`

⇒ `y = tanx - 1 + c.e^-tanx`