Walking the Slope
If we can't solve a DE with a formula, we can "walk" the solution step by step using the derivative as our compass. This is Euler's Method.
\[y_{new} = y_{old} + f(x_{old}, y_{old}) \cdot \Delta x\]
Worked Examples
Example 1: A Small Step
Solve \(\frac{dy}{dx} = x+y\) with \(y(0) = 1\) and a step size \(\Delta x = 0.1\).
- Start: \((0, 1)\). Slope = \(0+1 = 1\).
- Next y: \(1 + (1)(0.1) = 1.1\).
- Next point: \((0.1, 1.1)\). Slope = \(0.1+1.1 = 1.2\).
- Next y: \(1.1 + (1.2)(0.1) = 1.22\).
- Result: At \(x=0.2, y \approx 1.22\).
The Bridge to Quantum Mechanics
Euler's method is the simplest form of Time Evolution in computer simulations. In Quantum Mechanics, we use more advanced versions (like the Runge-Kutta method) to simulate how a wavefunction changes over time. When you see an animation of a quantum particle "splatting" against a wall, you are watching a computer solve a differential equation using millions of tiny steps, just like Euler's method.