Euler's Method
When exact solutions are impossible, we use computers. This is the simplest numerical method for DEs.
Introduction
Most real-world differential equations cannot be solved using the analytical tricks we've learned so far.
Euler's Method approximates the solution curve by stitching together many tiny tangent lines.
The Algorithm
Recipe
Given , and step size :
Think of it as: New Value = Old Value + (Step Size)(Slope).
Visual: Step Size Matters
Interactive: Euler vs Exact
For , Euler's method gives . This approximates and converges as .
Worked Examples
Example 1: Manual Calculation
Use Euler's Method with to estimate for , .
Step 0:
. Slope .
Step 1:
.
.
New Slope .
Step 2:
.
.
Result:
.
(Exact answer is ).
Example 2: Non-Linear Setup
Set up the first two steps for , with .
Step 1:
. Slope .
.
.
Step 2:
Slope .
.
.
So .
Example 3: Error Analysis
If and , does Euler's method overestimate or underestimate the solution for ?
1. Concavity:
.
Since and , the solution is positive and increasing.
So . The solution is Concave Up.
2. Tangent Lines:
Tangent lines lie below a concave up curve.
Since Euler's method follows tangent lines, it will consistently Underestimate the true value.
Practice Quiz
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