The general solution of ${x^2}\frac{{dy}}{{dx}} = 2$ is

The solution of the differential equation $x = 1 + xy\frac{dy}{dx} + \frac{(xy)^2}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{(xy)^3}{3!}\left(\frac{dy}{dx}\right)^3 + \dots$ is

The solution of the differential equation,$2x^2y \frac{dy}{dx} = \tan(x^2y^2) - 2xy^2$ given $y(1) = \sqrt{\frac{\pi}{2}}$ is

The solution of the differential equation $\sin \left( \frac{dy}{dx} \right) = a$ with the initial condition $y(0) = 1$ is:

The solution of $\frac{dy}{dx} = e^{-2x}$ with the condition $y(\log 2) = \frac{1}{16}$ is $y =$

The solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ is