I'm trying to solve the problem (x+y+1)^2 dy/dx + (x+y+1)^2 + x^3 = 0. I've tried the substituting for x+y+1, x+y and (x+y+1)^2 but all to no avail. Could somebody please give a suggestion? This problem apparently just requires a change of variables and separation.
Here is a link
Exact differential equation - Wikipedia, the free encyclopedia
This should also be in your book.
An equation is exact if there is a function such that
Where is the position vector.
Since the mixed partials match we know such a function exists so
But we also know what the partial derivative with respect to must be so we can solve for
This gives
is an implicit solution to the equation.
You said in your OP that this problem "apparently just requires a change of variables and separation." Does that mean you are required to solve it that way? Or could you solve it using the exact differential method that TheEmptySet has outlined for you? Because the exact differential method that TheEmptySet has outlined for you is not a substitution method, nor is it a separation of variables idea.