The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$ 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:

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

The solution of the given differential equation $\frac{dy}{dx} + 2xy = y$ is

Let $y=y(x)$ be the solution curve of the differential equation $(y^{2}-x) \frac{dy}{dx}=1$ satisfying $y(0)=1$. This curve intersects the $x$-axis at a point whose abscissa is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 3(\tan^2 x + 1)y = \sec^2 x$,with the initial condition $y(0) = \frac{1}{3} + e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to: