The curve passing through the point $(-1, 1)$, given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$ also passes through the point $\left( k, -\frac{1}{2} \right)$, then

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4 + \frac{d^4y}{dx^4} + \log\left(\frac{dy}{dx}\right) = 0$ is

The general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}+\text{cosec}\left(\frac{y}{x}\right)=0$ is

(Consider $\log x=\log_e x$)

The solution of the initial value problem $x\frac{dy}{dx} = 2, y(1) = 2$ is

For the differential equation $(x + y)dy + (x - y)dx = 0$, which of the following is/are correct?

(A) Differential equation is homogeneous

(B) Order of differential equation is 1

(C) Integrating factor of differential equation is $e^x$

(D) Degree of the equation is not defined

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