If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$, then the value of $m + n$ is:

Which of the following statements is/are true?

(A) $(\tan^{-1} y - x)dy = (1 + y^2)dx$ is a differential equation where variables are separable.

(B) $(1 + x^2)dy + 2xydy = \cot x \ dx (x \neq 0)$ is a first order linear differential equation.

(C) $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$ is not a homogeneous differential equation.

(D) $(xy)dx - (x + y^2)dy = 0$ is a homogeneous differential equation.

Choose the correct answer from the options given below:

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x+2}{3-y}, (y \neq 3), y(0) = 0$

represents

The particular solution of the differential equation $\sec^2x\tan y\,dx - \sec^2y\tan x\,dy = 0$, $y=\frac{\pi}{4}$ when $x=\frac{\pi}{4}$, is

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE?

(A) Product of order and degree is 1

(B) Integrating factor is $x^3$

(C) Integrating factor is $3x$

(D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant.

Choose the correct answer from the options given below:

Let $y = f(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx} + 2y^2 = 0$ if $f(1) = 1$, then $f(2)$ is equal to

Match List-I with List-II

Choose the correct answer from the options given below: