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:

We need to solve the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ given that the curve passes through $(0, 0)$.

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Factor the right-hand side by grouping:

$\frac{dy}{dx} = (1 + x) + y(1 + x) = (1 + x)(1 + y)$

This lets us separate variables — all $y$ terms on one side, all $x$ terms on the other.

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Separating variables:

$\frac{dy}{1 + y} = (1 + x)\, dx$

Integrating both sides:

$\int \frac{dy}{1 + y} = \int (1 + x)\, dx$

$\ln|1 + y| = x + \frac{x^2}{2} + C$

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Applying the initial condition $(0, 0)$:

$\ln|1 + 0| = 0 + 0 + C$

$\ln(1) = C \implies C = 0$

So the equation of the curve is:

$\ln(1 + y) = x + \frac{x^2}{2}$

Or equivalently:

$y = e^{x + x^2/2} - 1$

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Checking Option (A): Is it a straight line?

A straight line has the form $y = mx + c$. Our equation $y = e^{x + x^2/2} - 1$ is an exponential function, not a straight line. ❌

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Checking Option (B): Is it a parabola?

A parabola is a polynomial of degree 2. Our equation involves an exponential — it is not a polynomial at all. ❌

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Checking Option (C): Does it pass through $\left(-1,\; \frac{1}{\sqrt{e}} - 1\right)$?

Substitute $x = -1$:

$y = e^{(-1) + \frac{1}{2}} - 1$

$= e^{-1/2} - 1$

$= \frac{1}{\sqrt{e}} - 1$

This matches the given point. ✅

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Checking Option (D): Is the equation $xy(x+1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$?

This equation represents four straight lines: $x = 0$, $y = 0$, $x = -1$, and $y = \frac{1}{\sqrt{e}} - 1$. Our curve is a smooth exponential function, not a collection of lines. The fact that the curve passes through certain points does not mean its equation is the product of lines through those points. ❌

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Only Option (C) is correct.

$\boxed{C}$