Let’s consider the first order separable differential equation \(y^{\,\prime }=2xe^y\).

(a)
Here is an (approximate) direction field for this differential equation on
\([-2,2]\times [-2,2]\). Let’s draw (part of) the solution curve to this differential equation that passes through the point \((0,0)\). Let’s use our drawing to estimate the value of this solution when \(x=0.5\).
A direction field.
Figure 1: A direction field
(b)
Does this equation have any constant solutions?

(c)
Let’s write this equation the form \(h(y)y^{\,\prime }=g(x)\).
(d)
Let’s integrate both sides with respect to \(x\).
(e)
Now let’s solve this differential equation for its general solution, and determine the solution that passes through \((0,0)\).
(f)
How good was our estimate?
(g)
Let’s determine the interval \(I\) such that our solution is valid on \(I\), but not valid on any interval \(J\neq I\) that contains \(I\).