JEE Main · 2010 · 11 Apr 2010 · Offline · Differential Equations

Solution of the differential equation

cos x d y = y (sin x - y) d x, 0 < x < \frac{π}{2}

Answer: (D)

Solution

The equation

cos x d y = y (sin x - y) d x

rearranges to

\frac{d y}{d x} = y tan x - y^{2} sec x

, a Bernoulli equation. Dividing by

y^{2}

and substituting

t = \frac{1}{y}

, so

\frac{d t}{d x} = - \frac{1}{y ^{2}} \frac{d y}{d x}

, turns it into the linear equation

\frac{d t}{d x} + t tan x = sec x

. Its integrating factor is

e^{\int t a n x d x} = sec x

, giving

\frac{d}{d x} (t sec x) = sec^{2} x

, so

t sec x = tan x + c

. Replacing

t

\frac{1}{y}

\frac{sec x}{y} = tan x + c

, that is

sec x = (tan x + c) y

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option

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