Question

Solve:

dy/dx = 1 + x + y + xy

Answer given in the book is : log | 1 + y| = x + 1/2 x2 + C

Answer 1

$\frac{dy}{dx} = 1 + x + y + xy \\ \\ \: \: \: \: \: \: \: \: = 1 + x + y(1 + x) \\ \\ \: \: \: \: \: \: \: \: = (1 + x)(1 + y) \\ \\ \: \: \: \: \: \: \: \: \frac{dy}{1 + y} = (1 + x)dx \\ \\ \\ take \: both \: side \: intigration \\ \\ \\ \: \: \: \: \: \: \: \: \: intigration \frac{dy}{1 + y} = intigration (1 + x)dx \\ \\ \\ log(1 + y) = x + {x}^{2} \\ \\ \\ log(1 + y) = x(1 + x)$