For the differential equation $\left(x \log _{e} x\right) d y=\left(\log _{e} x-y\right) d x$ (A) Degree of the given differential equation is $1$. (B) It is a homogeneous differential equation. (C) Solution is $2y \log _{\mathrm{e}} \mathrm{x}+A=\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}$, where $A$ is an arbitrary constant (D) Solution is $2 y \log _{e} x+A=\log _{e}\left(\log _{e} x\right)$, where $A$ is an arbitrary constant Choose the correct answer from the options given below :

If m and n are respectively the order and degree of the differential equation :

$\left(\frac{d^2 y}{dx^2}\right)^5 + 6 \frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}} + \frac{d^3 y}{dx^3} = x^2 + 5$, then :

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

Match List-I with List-II

Choose the correct answer from the options given below:

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The general solution of $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$ is:

(given that $C$ is the constant of integration)

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