Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y) e^{2x} dx - (7+e^{2x}) dy = 0$,then $k$ is equal to

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

Let the solution curve of the differential equation $x dy - y dx = \sqrt{x^{2} + y^{2}} dx$, where $x > 0$ and $y(1) = 0$, be $y = y(x)$. Then $y(3)$ is equal to:

The solution of $\frac{dy}{dx} = \frac{x \log x^2 + x}{\sin y + y \cos y}$ is

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:

If $\frac{dy}{dx} = y + 3$ and $y(0) = 2$,then find the value of $y(\log 2)$.