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:

Which of the following differential equations are of order 2 and degree 1?

(A). $\frac{d}{dx}\left(\frac{dy}{dx}\right) + x\frac{dy}{dx} = y$

(B). $\frac{d^2y}{dx^2} + x = \sqrt{\frac{dy}{dx} - y}$

(C). $\left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} = \log x$

(D). $\left(\frac{dy}{dx}\right)^2 + \frac{dy}{{dx}} = 3$

Choose the correct answer from the options given below:

Match List-I with List-II

Choose the correct answer from the options given below:

The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:

Match the LIST-I with LIST-II

[$a$ is an arbitrary constant]

Choose the correct answer from the options given below:

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

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