 So let's see if we can try to solve an exact differential equation where we don't start with the level curve. So again, if we have an exact differential equation, our solutions correspond to the level curves of some function. The general solution will be a family of curves, and the particular solution will depend on the initial conditions. So let's see if we can solve this differential equation where y of 1 is equal to 0. So again, we start with the assumptions. Suppose the solutions correspond to the level curves f of x, y equals c for some function f. Then we can find the complete derivative, and since both of these are equal to 0, we can now compare our two equations. So comparing our two equations. On the left, the quantity in front of dy dx is our partial with respect to y. So on the right, the quantity in front of dy dx should be our partial with respect to y. So that gives us our partial of f with respect to y is x e to the power y. We can find f by finding the antiderivative with respect to y, and so that's going to be where because we are dealing with a partial derivative with respect to y, our constant is going to be some function of x. Similarly, our partial derivative of f with respect to x is going to be whatever is left over, and so that's going to be 1 over x plus e to the y. And again, we can find f by finding the antiderivative with respect to x, and that's going to be where we include some function of y only. So now let's compare our two expressions for f of x, y. On the one hand, it's x e to the y plus some function of x. On the other hand, it's log x plus x e to the y plus some function of y. And if we rearrange and simplify, we get... And if we look at this, that says that c1 of x must be log of x, and c2 of y must be a constant, which we can now ignore because that will become part of our level curve equation. So we could note that f of x, y is log x plus x e to the y, but remember, your solution should not include any new functions or variables. And so while we do have everything here by writing down this assumption that we're looking at the level curves f of x, y equals c, and then giving what f of x, y is, it's better to make that explicit. Our function, log x plus x e to the y, is equal to c. And finally, we do have an initial condition. We have y of 1 is equal to 0. So to find the particular solution, we'll incorporate the initial condition y of 1 equals 0. So remember that this says that our x value is 1 and our y value is 0. So if we substitute these into our equation, we get c equal to 1. And so c equal to 1 gives us the solution satisfying our initial condition. Now we do have y defined implicitly as a function of x. In some cases it might be possible to express y as an explicit function of x. So this equation we could actually solve for y if we wanted to. But in most cases it's not worth the effort of trying to solve the equation for y. And that means we can leave our solution in this form.