The differential equation $\frac{dy}{dx} = \frac{1}{ax + by + c}$,where $a, b, c$ are all non-zero real numbers,is

Let $f: R \rightarrow R$ be a continuous function satisfying $f(x)=x+\int_0^x f(t) dt$,for all $x \in R$. Then,the number of elements in the set $S=\{x \in R: f(x)=0\}$ is

If $\cos x \frac{dy}{dx} - y \sin x = 6 x$,$0 < x < \frac{\pi}{2}$,then the general solution of the differential equation is

$A$ function $f(x)$ satisfies the condition $f(x) = f'(x) + f''(x) + f'''(x) + \dots \infty$,where $f(x)$ is an indefinitely differentiable function and the dash denotes the order of the derivative. If $f(0) = 1$,then $f(x)$ is:

Find a particular solution satisfying the given condition: $\frac{dy}{dx} - 3y \cot x = \sin 2x$; $y = 2$ when $x = \frac{\pi}{2}$.

If $x \phi(x) = \int_{5}^{x} (3t^{2} - 2 \phi'(t)) dt$,$x > -2$,and $\phi(0) = 4$,then $\phi(2)$ is .... .