The solution of the differential equation,$(x + 2y^3) \frac{dy}{dx} = y$ is :

The solution of the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is

The integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

The equation of the curve passing through $(1,2)$ and whose tangent at any point $(x, y)$ makes an angle $\tan ^{-1}(2 x+3 y)$ with the $X$-axis is .........

If $y = y(x)$,$x \in \left(0, \frac{\pi}{2}\right)$ is the solution curve of the differential equation $(\sin^2 2x) \frac{dy}{dx} + (8 \sin^2 2x + 2 \sin 4x) y = 2 e^{-4x} (2 \sin 2x + \cos 2x)$,with $y\left(\frac{\pi}{4}\right) = e^{-\pi}$,then $y\left(\frac{\pi}{6}\right)$ is equal to: