Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration (D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, where C is constant of integration (E) If $y(1) = 1$, then its particular solution is $y = x$ Choose the correct answer from the options given below:

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is

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

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$, is :

General solution of the differential equation $\dfrac{dy}{dx} = e^{ax+by}$ is: (where C is an arbitrary constant)

Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is: