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

The integrating factor of the differential equation $\frac{dy}{dx}(1+x) - xy = 1-x$ is . . . . . . .

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 $.....$

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} = \frac{y}{x}(1 + xy^2(1 + \log_e x))$ for $x > 0$ and $y(1) = 3$. Then $\frac{y^2(x)}{9}$ is equal to:

If the solution curve of the differential equation $(y-2 \ln x) dx + (x \ln x^2) dy = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^4, \alpha)$,then $\alpha$ is equal to $................$.

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