... Abuse of notation perhaps.
Now the hard part: isolate y.
No, that's incorrect. x is not a constant and cannot be treated as one in the first integral.
farlist, that is NOT "a pretty easy question" but here is what I came up with: . Just because it looked like it might simplify things, I let . Then so that and . Putting that into the equation: becomes . That's a separable equation and should be easy to solve.
We can make exact via the integrating factor . Check the book on the form. This yields:
Then the solution is
Solving for we get
The solution is then:
in which can be solved for explicitly (how?).
And this is how I checked the solution in Mathematica:
Code:n[7]:= ysol = First[Derivative[1][y][x] /. Solve[D[x*Exp[y[x]^2] - (1/2)*Exp[y[x]^2]*(y[x]^2 - 1), x] == 0, Derivative[1][y][x]]] Simplify[(2*x*y[x] - y[x]^3)*ysol + 1] Out[7]= 1/(y[x]*(-2*x + y[x]^2)) Out[8]= 0