The integrating factor of the differential equation $3 x \log _{e} x \frac{d y}{d x}+y=2 \log _{e} x$ is given by

Let $y=y(x)$ be the solution curve of the differential equation $\sin(2x^2) \ln(\tan x^2) dy + (4xy - 4\sqrt{2}x \sin(x^2 - \frac{\pi}{4})) dx = 0$ for $0 < x < \sqrt{\frac{\pi}{2}}$,which passes through the point $(\sqrt{\frac{\pi}{6}}, 1)$. Then $|y(\sqrt{\frac{\pi}{3}})|$ is equal to $.....$

The solution of the differential equation $\frac{dy}{dx} + 2y = e^{-x}$ is

Let the solution curve $y = y(x)$ of the differential equation $\frac{dy}{dx} - \frac{3x^5 \tan^{-1}(x^3)}{(1+x^6)^{3/2}} y = 2x \exp \left( \frac{x^3 - \tan^{-1}(x^3)}{\sqrt{1+x^6}} \right)$ pass through the origin. Then $y(1)$ is equal to:

The solution of the differential equation $\frac{dy}{dx} + x \sin^2 y = \sin y \cos y$ is:

$y+x^2=\frac{dy}{dx}$ has the solution