The general solution of the differential equation $\frac{dx}{dt} = \frac{x \log x}{t}$ is

The general solution of $\frac{dy}{dx} = \frac{x^3(y^4+1)}{\left[2y^{-2/3} + 3\left(\frac{x}{y^{1/3}}\right)^2\right]^{3/2}}$ is

Find a particular solution satisfying the given condition:
$(x^{3}+x^{2}+x+1) \frac{dy}{dx} = 2x^{2}+x; y=1$ when $x=0$

If $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$, then $|f(xy)|$ is equal to

The particular solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ given that $y = e^2$ when $x = e$ is:

Let $y = y(x)$ be the solution curve of the differential equation $(1 + \sin x) \frac{dy}{dx} + (y + 1) \cos x = 0$ with the condition $y(0) = 0$. If the curve $y = y(x)$ passes through the point $(\alpha, -\frac{1}{2})$, then a value of $\alpha$ is: