Introduction: Why Second-Order DEs Need a Special Approach
In AHL 5.17 you applied Euler's method to first-order differential equations of the form . Many real physical systems — springs, pendulums, electrical circuits — are governed by second-order DEs, where the highest derivative is .
Euler's method cannot be applied directly to a second-order DE, because the iterative formula only steps one derivative at a time. The strategy is to convert the second-order DE into a system of two coupled first-order DEs, and then apply Euler's method to both simultaneously.
Think of it like tracking a moving car. Position and velocity are two separate quantities that both change each time step — you update them together, each depending on the other. A second-order DE is just saying: "the acceleration depends on position and velocity."
Converting a Second-Order DE into a System
Consider a general second-order DE:
The conversion procedure is always the same two-step process:
Step 1 — Introduce a new variable for the first derivative:
Step 2 — Rewrite as a system of two first-order DEs:
Now instead of one equation involving , we have two equations each involving only a first derivative. This is exactly the format Euler's method can handle.
Initial conditions for a second-order DE always specify both and . In the new system, these become and — the two starting values you need.
Never try to apply Euler's method directly to the second-order form. You must perform the substitution first — this is the key step that unlocks the whole method.