The particular solution of $e^{\frac{y}{x}} = x, y(1) = 3, x > 0$ is . . . . . . .

If $y=f(x)$ is the solution of the differential equation $(1+\cos^2 x) f'(x) - f(x) \sin 2x = 4 \sin 2x$ with $f(0)=0$,then $f(\frac{\pi}{3})=$

Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{dy}{dx} + 2x \sin y = x^3 \cos y$,with the initial condition $y(1) = 0$. Then the value of $y(\sqrt{3})$ is equal to:

The particular solution of the differential equation $\sin^{2} y \frac{dx}{dy} + x = \cot y$ when $x = 0$ and $y = \frac{3\pi}{4}$ is

The solution of $\cos y + (x \sin y - 1) \frac{dy}{dx} = 0$ is

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