The order of $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left[a \frac{d^2y}{dx^2}\right]^{\frac{1}{2}}$ is

The solution of the differential equation $(x+1)\frac{dy}{dx} = 1 + y$ is

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

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:

Let $y(x) = a(x + 1) \log(x + 1) + bx + 5$ be the solution of the differential equation e$^\frac{dy}{dx} = x + 1_{;}y(0) = 5$, then the value of $(a + b)$ is:

Let $e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0$, where $C \in \mathbb{R}$ be a particular solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^ydx = 0$ and passes through the point $(1, 1)$. The value of $(\alpha + \beta + \gamma + \delta - C)$ is

The integrating factor of the differential equation $y\log_e(10y).\dfrac{dx}{dy}+x=\dfrac{1}{[\log_e(10y)]^2}$, is