Question

Solution of the differential equation \( xy \frac{dy}{dx} = 1 + x + y + xy \) is

• A. \( \log(x(1+y)) = c \)
• B. \( (y - x) \log(x(1 + y)) = c \)
• C. \( (y + x) \log(x) = c \)
• D. \( \log(x(1 + y)) = c \)

Hint:

To solve such differential equations, express the equation in terms of a separable form and integrate both sides.

Answer 1

The Correct Option is A

We are given the differential equation:

\[ xy \frac{dy}{dx} = 1 + x + y + xy \]

Rearranging the terms to separate variables:

\[ xy \frac{dy}{dx} = (1 + x) + y(1 + x) \]

Now, simplifying the equation:

\[ \frac{dy}{dx} = \frac{(1 + x) + y(1 + x)}{xy} \]

We can factor out \( (1 + x) \) from the numerator:

\[ \frac{dy}{dx} = \frac{(1 + x)(1 + y)}{xy} \]

Separating the variables \( x \) and \( y \), we get:

\[ \frac{dy}{1 + y} = \frac{(1 + x)}{x} \cdot \frac{dx}{y} \]

Integrating both sides, we get the solution:

\[ \log(x(1 + y)) = c \]

This matches option (A).