The solution of the differential equation dydx=1xyx2sin ⁡y2+1 is

The given differential equation can be written as dxdy=xyx2sin⁡ y2+1 ⇒1x3dxdy−1x2y=ysin ⁡y2This equation is reducible to linear equation, so putting −1/x2=u, the last equation can be written as dudy+2uy=2ysin⁡y2The integrating factor of this equation is ey2 So required solution is⇒uey2=∫2ysin⁡y2⋅ey2dy+C⇒uey2=12ey2sin⁡y2−cos⁡y2+C⇒ 2u=sin⁡y2−cos⁡y2+Ce−y2⇒ 2=x2cos⁡y2−sin⁡y2−2Ce−y2.

The solution of the differential equation dydx=1xyx2sin ⁡y2+1 is

The given differential equation can be written as dxdy=xyx2sin⁡ y2+1 ⇒1x3dxdy−1x2y=ysin ⁡y2This equation is reducible to linear equation, so putting −1/x2=u, the last equation can be written as dudy+2uy=2ysin⁡y2The integrating factor of this equation is ey2 So required solution is⇒uey2=∫2ysin⁡y2⋅ey2dy+C⇒uey2=12ey2sin⁡y2−cos⁡y2+C⇒ 2u=sin⁡y2−cos⁡y2+Ce−y2⇒ 2=x2cos⁡y2−sin⁡y2−2Ce−y2.

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The solution of the differential equation $\frac{d y}{d x} = \frac{1}{x y [x^{2} \sin y^{2} + 1]}$ is

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$x^{2} (\cos y^{2} - \sin y^{2} - 2 c e^{- y^{2}}) = 2$

$x^{2} (\cos y^{2} - \sin y^{2} - e^{- y^{2}}) = 4$

$x^{2} (\cos y^{2} - \sin y^{2}) = 2 c e^{- y / 2}$

$y^{2} (\cos x^{2} - \sin y^{2} - 2 c e^{- y^{2}}) = 2$

answer is A.

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Detailed Solution

The given differential equation can be written as

$\frac{d x}{d y} = x y [x^{2} \sin y^{2} + 1] \Rightarrow \frac{1}{x^{3}} \frac{d x}{d y} - \frac{1}{x^{2}} y = y \sin y^{2}$

This equation is reducible to linear equation, so putting $- 1 / x^{2} = u$ , the last equation can be written as $\frac{d u}{d y} + 2 u y = 2 y \sin y^{2}$

The integrating factor of this equation is $e^{y^{2}}$ So required solution is

$\begin{aligned} \Rightarrow u e^{y^{2}} = \int 2 y \sin y^{2} \cdot e^{y^{2}} d y + C \\ \Rightarrow u e^{y^{2}} = \frac{1}{2} e^{y^{2}} (\sin y^{2} - \cos y^{2}) + C \end{aligned}$

$\begin{array}{l} \Rightarrow 2 u = (\sin y^{2} - \cos y^{2}) + C e^{- y^{2}} \\ \Rightarrow 2 = x^{2} [\cos y^{2} - \sin y^{2} - 2 C e^{- y^{2}}] . \end{array}$