Choose the correct answer from the options given below: | List-I | List-II | | --- | --- | | (A) Integrating factor of $ x \, dy - (y + 2x^2) \, dx = 0 $ | (I) $ \frac{1}{x} $ | | (B) Integrating factor of $ (2x^2 - 3y) \, dx = x \, dy $ | (II) $ x $ | | (C) Integrating factor of $ (2y + 3x^2) \, dx + x \, dy = 0 $ | (III) $ x^2 $ | | (D) Integrating factor of $ 2x \, dy + (3x^3 + 2y) \, dx = 0 $ | (IV) $ x^3 $ |

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

The particular solution of the differential equation $\sec^2x\tan y\,dx - \sec^2y\tan x\,dy = 0$, $y=\frac{\pi}{4}$ when $x=\frac{\pi}{4}$, is

For the differential equation $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = k, k \neq 0$, which one of the following is TRUE ?

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 differential equation whose order and degree are 3 and 2 respectively is

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 :