For the differential equation $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = k, k \neq 0$, which one of the following is TRUE ?

Let the degree and order of the differential equation $2x^3\frac{dy}{dx}-5\left(\frac{d^2y}{dx^2}\right)^2=6\left(\frac{dy}{dx}\right)^3$ be $m$ and $n$ respectively. Then

(A) m = 2

(B) n = 3

(C) m = 3

(D) mn = 4

Choose the correct answer from the options given below:

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$

(A) Integrating factor is $2x$

(B) Integrating factor is $x^2$

(C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant.

(D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant.

Choose the correct answer from the options given below:

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

The order and degree of the differential equation $\left[2+\left(\dfrac{dy}{dx}\right)\right]^{5/2}=7\dfrac{d^3y}{dx^3}$ is

General solution of the differential equation $x(xdy - ydx) = y\,dx$ is :(where c is an arbitrary constant)

The general solution of the differential equation $x\,dx = (x^2+1)\,dy$ is (where C is an arbitrary constant)