Slope Fields, Equilibrium Solutions, and Concavity

Reading a differential equation before solving it

A slope field draws, at each lattice point $(x,y)$, a short segment of slope $\frac{dy}{dx}$ evaluated there. A solution curve is a curve that is everywhere tangent to those segments; one initial condition picks out one curve.

Three readings that AP asks for again and again:

• Segments identical along every horizontal line means $\frac{dy}{dx}$ depends on $y$ alone; identical along every vertical line means it depends on $x$ alone.
• A horizontal row of segments at $y=c$ signals the equilibrium solution $y=c$. Verify it by substitution: the constant function makes both sides zero.
• By uniqueness of solutions of an initial value problem, two distinct solution curves never cross. An equilibrium solution therefore acts as a wall.

To decide concavity without solving, differentiate the equation itself. If $\frac{dy}{dx}=f(x,y)$, then $\frac{d^{2}y}{dx^{2}}=f_x+f_y\cdot\frac{dy}{dx}$ in practice: differentiate the right-hand side with respect to $x$, remembering that $y$ is a function of $x$.