The particular solution of the differential equation $\log \left(\frac{d y}{d x}\right)=x$, when $x=0, y=1$ is ..............

The p.d.f of a random variable $x$ is given by

$$\begin{aligned} & f(x)=\frac{1}{4 a}, \quad 00) \\ & =0 \text {, otherwise } \end{aligned}$$

and $P\left(x<\frac{3 a}{2}\right)=k P\left(x>\frac{5 a}{2}\right)$ then $k=$ ..............

If the function $f(x)=\frac{\left(e^{k x}-1\right) \tan k x}{4 x^2}, x \neq 0$

$$\qquad \qquad=16 \qquad x=0$$

is continuous at $x=0$, then $k=\ldots \ldots$

The solution of the differential equation $y d x-x d y=x y d x$ is ......