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

The solution of the differential equation $\frac{dy}{dx} = (x - y)^2$ when $y(1) = 1$ is:

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$ with $y(0)=3$ is $\alpha x+\beta y+3 \log_e|2x+3y-\gamma|=6$,then $\alpha+2\beta+3\gamma$ is equal to

Let $y=f(x)$ be the solution of the differential equation $y(x+1) dx - x^2 dy = 0$ with the initial condition $y(1)=e$. Then $\lim _{x \rightarrow 0^{+}} f(x)$ is equal to

Let $f: R \rightarrow R$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $\frac{dy}{dx}=(2+5y)(5y-2)$,then the value of $\lim_{x \rightarrow -\infty} f(x)$ is:

The solution of the differential equation $(2x - 4y + 3) \frac{dy}{dx} + (x - 2y + 1) = 0$ is ($C$ is an arbitrary constant).