# Physics 212, 2018: Lectures 7

Let's focus for now on a system of a single differential equation, like the constrained growth problem. If the system settled down, then in the equation ${\displaystyle dx/dt=f(x,a)}$ the time derivative is zero, and the equation becomes ${\displaystyle f(x,a)=0}$. It's an algebraic equation! How do we find its roots? We use the Newton-Raphson method, which I will derive in class.
Your own work: use the scripts I provided and change them to solve a quadratic equation. Compare the solution to the output of a function that gives an exact solution, which you wrote last time. Verify that different roots can be obtained using different initial conditions. Then use the scripts to solve an equation ${\displaystyle e^{x}-ax=0}$ for arbitrary ${\displaystyle a}$