Question

Find the general solution (implicit if necessary, explicit if convenient) to the differential equation

y ′ = 1+x+y+xy.

Primes denote the derivatives with respect to x. ( Suggestion: Factor the right-hand side.)

Answer 1

We can factor the right-hand side of the equation above to get:

y ′ = (1 + x)(1 + y).

In this form it is obviously a separable differential equation, and we can rewrite it as:

If we take the antiderivative of both side we get:

Exponentiating both sides we get:

So, our final solution is: