Consider the differential equation $\frac{dy}{dx} = \frac{y+1}{x+1}$, and $y=0$ when $x=2$. The value of $y$ at $x=3$ is :

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

[Where $C$ is an arbitrary constant]

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE?

(A) Product of order and degree is 1

(B) Integrating factor is $x^3$

(C) Integrating factor is $3x$

(D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant.

Choose the correct answer from the options given below:

If $\dfrac{dy}{dx} = e^{x-y} + x^2e^{-y}$, then the solution of the differential equation will be:-

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:

If the order and degree of the differential equation $y + \dfrac{dy}{dx} = \int y\,dx$ are m and n respectively, then (m - n) is equal to

If the slope of the tangent to the curve $y = y(x)$ at any point $(x, y)$ is $\frac{2x}{y^2}$ and the curve passes through the point $\left(\frac{1}{\sqrt3}, 1\right)$, then equation of curve is