The solution of the differential equation $\frac{dy}{dx} = \frac{\lambda^2}{(x+y)^2}$ ($\lambda$ is constant) is:

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. If it passes through the point $(0,0)$, then which of the following is/are true?

(A) it is a straight line.

(B) it is a parabola.

(C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$

(D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$

Choose the correct answer from the options given below:

Choose the correct statements:

A. The order and degree (if defined) of a differential equation are always positive integrals

B. The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation

C. If $\frac{dy}{dx} + P(x)y = Q(x)$ then Integrating factor $= e^{\int P(x)dx}$

D. The sum of order and degree of differential equation $1 + (y'')^5 = (y''')^3$ is $8$

E. If the solution of a differential equation of order $n$, contains $n$ arbitrary constant, then it is called a general solution

Choose the correct answer from the options given below:

The solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 5x + 2y$ is given by

The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$

The solution of the differential equation $9+y^2\dfrac{dy}{dx}=x^2$ is :

[Where $C$ is an arbitrary constant]

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