If y = y ( x ) is the solution of the differential equation d y d x = ( t a n x - y ) s e c 2 x , x ∈ - π 2 , π 2 , such that y 0 = 0 , then y - π 4 is equal to:

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TestHub · Accepted Answer

The given differential equation can be written as
$\frac{d y}{d x} + y {s e c}^{2} x = t a n x . {s e c}^{2} x$
Integrating factor $= e^{\int {s e c}^{2} x d x} = e^{t a n x}$
Hence, solution of given differential equation,
$y . e^{t a n x} = \int t a n x . {s e c}^{2} x . e^{t a n x} d x$
$\Rightarrow y . e^{t a n x} = t a n x . e^{t a n x} - \int s e c^{2} x . e^{t a n x} d x$ (using integration by parts)

$\Rightarrow y . e^{\tan x} = \tan x . e^{\tan x} - e^{\tan x} + c$

Given, $y (0) = 0 \Rightarrow c = 1$

Solution of given differential is

$y . e^{t a n x} = t a n x . e^{t a n x} - e^{t a n x} + 1$

Hence, $y (- \frac{π}{4}) = \frac{(- 1) e^{- 1} - e^{- 1} + 1}{e^{- 1}}$

$\Rightarrow y (- \frac{π}{4}) = e - 2$

Mathematics - Differential Equation Question with Solution | TestHub

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