Solution of the differential equation 2 y sin x dy - dx -2 sin x cos x - y 2cos x satisfyingy - - 2-1 is given byA- y2-cos x-1B- y 2-sin x -4 cos 2 xC- y 2-sin xD- y -sin 2 x

The correct option is C 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 = ( 1/2) cos ...

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Standard XII

Mathematics

General Solution of a Differential Equation

Solution of t...

Question

Solution of the differential equation $2 y s i n x \frac{d y}{d x} = 2 s i n x c o s x - y^{2} c o s x s a t i s f y i n g y (π / 2) = 1$ 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 $2 y s i n x \frac{d y}{d x} + y^{2} c o s x = s i n 2 x \Rightarrow \frac{d}{d x} (y^{2} s i n x) = s i n 2 x \Rightarrow y^{2} s i n x = (- 1 / 2) c o s 2 x + C . S o (y (π / 2))^{2} s i n (π / 2) = (- 1 / 2) c o s (2 π / 2) + C \Rightarrow C = 1 / 2. H e n c e y^{2} s i n x = (1 / 2) (1 - c o s 2 x) = s i n^{2} x \Rightarrow y^{2} = s i n x$

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Similar questions

Solution of the differential equation $2 y sin x \frac{d y}{d x} = 2 sin x cos x - y^{2} cos x s a t i s f y i n g, y (\frac{π}{2}) = 1$

is given by?

Q. Solution of

2 y sin x \frac{d y}{d x} = 2 sin x \cdot cos x - y^{2} cos x, x = \frac{π}{2}, y = 1

is given by

Q. For each of the differential equations, find the general solution:
1.

\frac{d y}{d x} = \frac{1 - c o s x}{1 + c o s x}

\frac{d y}{d x} = \sqrt{4 - y^{2}} (- 2 < y < 2)

Q. The solution of the differential equation

\frac{d y}{d x} + \frac{y}{2} sec x = \frac{tan x}{2 y},

where

0 \leq x \leq \frac{π}{2},

and

y (0) = 1,

is given by:

Q. The solution of the differential equation

\frac{d y}{d x} = 1 + x + y^{2} + x y^{2}, y (0) = 0

is
(a)

y^{2} = \exp (x + \frac{x^{2}}{2}) - 1

(b)

y^{2} = 1 + C \exp (x + \frac{x^{2}}{2})

y = \tan (x + \frac{x^{2}}{2})

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

The correct option is A y2=sin xThe given equation can be written as 2y sin xdydx+y2 cos x=sin 2x⇒ddx(y2 sin x)=sin 2x⇒y2 sin x=(−1/2) cos 2x+C.So (y(π/2))2 sin (π/2)=(−1/2)cos(2π/2)+C⇒C=1/2.Hence y2 sin x=(1/2)(1−cos 2x)=sin2x⇒y2=sin x

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