The solution of dy/dx = 1 + x + y + xy is

The correct option is: log (1 + y) = x + x²/2 + k

To solve the given differential equation, we separate variables and integrate as follows:

Given equation:
$$\frac{dy}{dx}=1+x+y+xy$$

Factorizing the right-hand side by grouping terms:
$$\frac{dy}{dx}=(1+x)+y(1+x)$$
$$\frac{dy}{dx}=(1+x)(1+y)$$

Separating the variables x and y:
$$\frac{dy}{1+y}=(1+x)dx$$

Integrating both sides of the equation:
$$\int \frac{dy}{1+y}=\int (1+x)dx$$

Evaluating the integrals gives:
$$\log (1+y)=x+\frac{x^2}{2}+k$$
where k is the integration constant.