Which of the following is a homogeneous differential equation ?

The solution of the differential equation $xdy - ydx = 0$ represents

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

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

Solution of differential equation $x\,dy - y\,dx = 0$ represents:

Which of the following are first order linear differential equations?

(A) $\frac{dx}{dy} + P_1(y)x = Q_1(y)$ : $P_1(y)$ and $Q_1(y)$ are functions of y or constant functions

(B) $\frac{dy}{dx} + P_2(x)y = Q_2(x)$ : $P_2(x)$ and $Q_2(x)$ are functions of x or constant functions

(C) $(x + y)\frac{dy}{dx} = x - 2y$

(D) $(1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3$

Choose the correct answer from the options given below:

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :