In the previous chapter we solved for the landing spot of a projectile by finding the root of the parabolic function derived from the equation of motion of an object in a constant gravitational field.

But what if we include other effects? Air resistance, for example, adds a velocity dependent term to the ODE. In such cases, there may not exist nice analytical formulas on which we can run our root finding algorithms. In such cases we typically will solve the equation numerically with a method such as the Euler or Predictor-Correction methods discussed in Chapter 2 : Physics.

However, those methods do not provide very reliable, quick answers. In this chapter we look at some improved ODE solving methods. These include the

• Midpoint or 2nd order Runge-Kutta method - this method obtains a better estimate for the increment in the function by using the slope at the center of the interval. Demo 1.
  
  
• 4th order Runge-Kutta method - this workhorse method provides a somewhat better estimate of the function increment than the 2nd order version by using a weighted average of four slopes from the sides and midpoint of the the interval. Demo 2.
  

In Chapter 2: Physics we examined initial value problems. Here we discuss Boundary Value Problems and the Shooting Method for solving such problems.