The integrating factor of the differential equation $x \frac{dy}{dx} - y = x^3, (x > 0)$ is . . . . . . .

Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-\sin(2y)=x^{3}(2-x^{3})\cos^{2}y,$ for $x\ne0.$ If $y(2)=0,$ then $\tan(y(1))$ is equal to

$A$ function $y=f(x)$ satisfies $f(x) \sin 2x + \sin x - (1 + \cos^2 x) f'(x) = 0$ with the condition $f(0) = 0$. Then $f(\frac{\pi}{2})$ is equal to

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 equation of the curve passing through the point $(0, \pi)$ and satisfying the differential equation $y dx = (x + y^3 \cos y) dy$ is

Let $x = x(y)$ be the solution of the differential equation $2y^2 \frac{dx}{dy} - 2xy + x^2 = 0, y > 1, x(e) = e$. Then $x(e^2)$ is equal to: