The solution of y3cos⁡xdx−xe1/y2dy=0 is

∫cos⁡xxdx=−12∫e1/y2−2/y3dy⇒2sin⁡x=−12e1/y2+c⇒e1/y2+4sin⁡x=c

The solution of y3cos⁡xdx−xe1/y2dy=0 is

∫cos⁡xxdx=−12∫e1/y2−2/y3dy⇒2sin⁡x=−12e1/y2+c⇒e1/y2+4sin⁡x=c

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$The solution of y^{3} \cos \sqrt{x} dx - \sqrt{x} e^{1 / y^{2}} dy = 0 is$

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$e^{1 / y^{2}} + 2 \sin \sqrt{x} = c$

$e^{{1/y}^{2}} +4sin \sqrt{x} = c$

$e^{{1/y}^{2}} -2sin \sqrt{x} = c$

$e^{{1/y}^{2}} -4sin \sqrt{x} = c$

answer is B.

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Detailed Solution

$\begin{array}{l} \int \frac{\cos \sqrt{x}}{\sqrt{x}} dx = - \frac{1}{2} \int e^{1 / y^{2}} (- 2 / y^{3}) dy \\ \Rightarrow 2 \sin \sqrt{x} = - \frac{1}{2} e^{1 / y^{2}} + c \Rightarrow e^{1 / y^{2}} + 4 \sin \sqrt{x} = c \end{array}$

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$The solution of y^{3} \cos \sqrt{x} dx - \sqrt{x} e^{1 / y^{2}} dy = 0 is$