Solution of the differential equation 2 y sin x dy - dx -2 sin x cos x - y 2cos x satisfying y -2-1 is given byA- y 2-sin xB- y-sin 2 xC- y2-cos x-1D- y2-sin x-4 cos 2 x

The correct option is A y^2 = sin xThe given equation can be written as 2y sin x dy/dx + y^2 cos x = sin 2x ⇒d/dx (y^2 sin x) = sin 2x ⇒ y^2 sin x = ( ...

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Standard XII

Mathematics

Particular Solution of a Differential Equation

Solution of t...

Question

Solution of the differential equation $2 y s i n x \frac{d y}{d x} = 2 s i n x c o s x - y^{2} c o s x$ satisfying $y (\frac{π}{2}) = 1)$ is given by

y^{2} = s i n x

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y = s i n^{2} x

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y^{2} = c o s x + 1

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y^{2} = s i n x = 4 c o s^{2} x

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Solution

The correct option is A $y^{2} = s i n x$
The given equation can be written as $2 y s i n x \frac{d y}{d x} + y^{2} c o s x = s i n 2 x$
$\Rightarrow \frac{d}{d x} (y^{2} s i n x) = s i n 2 x \Rightarrow y^{2} s i n x = (- \frac{1}{2}) c o s 2 x + C .$
So $(y (\frac{π}{2}))^{2} s i n (\frac{π}{2}) = (- \frac{1}{2}) c o s (\frac{2 π}{2}) + C \Rightarrow C = \frac{1}{2}$
Hence $y^{2} s i n x = (\frac{1}{2}) (1 - c o s 2 x) = s i n^{2} x \Rightarrow y^{2} = s i n x$

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Solution of the differential equation 2y sin xdydx=2 sin x cos x−y2 cos x satisfying y(π2)=1)is given by

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