Dy/dx(x+y)=x^2+xy+x+1?

Question

USING THE SUBSTITUTION : Y=V-X

Answer 1

On the left, is that supposed to be dy/dx OF x+y?  Or is it (dy/dx) TIMES (x+y)?

Answer 2

y = v - x where v is itself a function of x.

dy/dx = dv/dx - 1

Substituting this in the original diff. eqn. gives

v(dv/dx - 1) = x^2 + x(v - x) + x + 1

v*(dv/dx) = v + vx + x + 1 = (v + 1)(x + 1)

This means that it is now a variables separable diff. eqn.

v*dv/(v + 1) = (x + 1)dx

Can you take it from there?

I don't think that you can get y as an explicit function of x. You will have to leave it as a complicated x, y relationship.

Answer 3

P(x) = -2 x and Q(x) = x