UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Euler's Method
First-Order Tangent Stepping

Euler's Method is the foundational numerical procedure for initial value problems $y' = f(x, y)$ . Like walking in a foggy landscape, it looks at the current slope $f(x_n, y_n)$ and takes a straight linear step of size $h$ into the future!

EULER FIRST-ORDER STEP RECURRENCETANGENT LINE APPRX

y_{n+1} = y_n + h \cdot f(x_n, y_n)

1. Current State (yₙ)Current solution value at coordinate

x_n

2. Local Slope f(xₙ, yₙ)Evaluated tangent rate of change

\frac{dy}{dx}

3. Step Size (h)Uniform horizontal increment

x_{n+1} = x_n + h

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the differential equation:

\frac{dy}{dx} = x + y, \quad \text{with initial condition } y(0) = 1

Find the value of $y(0.2)$ using step size $h = 0.1$ (2 steps)!

STEP 01 • FIRST TANGENT STEP (x = 0 → x = 0.1)

At $x_0 = 0, y_0 = 1$ , compute starting slope $f(0, 1)$ :

f(x_0, y_0) = 0 + 1 = 1.0

y_1 = y_0 + h \cdot f(x_0, y_0) = 1.0 + (0.1)(1.0) = 1.1000

STEP 02 • SECOND TANGENT STEP (x = 0.1 → x = 0.2)

Now at updated point $x_1 = 0.1, y_1 = 1.1$ , compute new slope $f(0.1, 1.1)$ :

f(x_1, y_1) = 0.1 + 1.1 = 1.2

y_2 = y_1 + h \cdot f(x_1, y_1) = 1.1 + (0.1)(1.2) = 1.2200

FINAL RESULT

Approximated Value: y(0.2) = 1.22000

\boxed{y(0.2) \approx 1.22000} \quad (\text{Exact: } 1.24281)

With step size $h = 0.1$ , Euler's Method predicts 1.22000 (compared to exact value $1.24281$ ).

LABORATORY

Interactive Live Calculator & Visualizer

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATION LABORATORY

Euler's Method Engine

PreviousTaylor's Series Method

Unit 4•19 of 28

NextModified Euler's Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Euler's Method
First-Order Tangent Stepping

EULER FIRST-ORDER STEP RECURRENCETANGENT LINE APPRX

y_{n+1} = y_n + h \cdot f(x_n, y_n)

1. Current State (yₙ)Current solution value at coordinate

x_n

2. Local Slope f(xₙ, yₙ)Evaluated tangent rate of change

\frac{dy}{dx}

3. Step Size (h)Uniform horizontal increment

x_{n+1} = x_n + h

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Given the differential equation:

\frac{dy}{dx} = x + y, \quad \text{with initial condition } y(0) = 1

Find the value of $y(0.2)$ using step size $h = 0.1$ (2 steps)!

STEP 01 • FIRST TANGENT STEP (x = 0 → x = 0.1)

At $x_0 = 0, y_0 = 1$ , compute starting slope $f(0, 1)$ :

f(x_0, y_0) = 0 + 1 = 1.0

y_1 = y_0 + h \cdot f(x_0, y_0) = 1.0 + (0.1)(1.0) = 1.1000

STEP 02 • SECOND TANGENT STEP (x = 0.1 → x = 0.2)

Now at updated point $x_1 = 0.1, y_1 = 1.1$ , compute new slope $f(0.1, 1.1)$ :

f(x_1, y_1) = 0.1 + 1.1 = 1.2

y_2 = y_1 + h \cdot f(x_1, y_1) = 1.1 + (0.1)(1.2) = 1.2200

FINAL RESULT

Approximated Value: y(0.2) = 1.22000

\boxed{y(0.2) \approx 1.22000} \quad (\text{Exact: } 1.24281)

With step size $h = 0.1$ , Euler's Method predicts 1.22000 (compared to exact value $1.24281$ ).

LABORATORY

Interactive Live Calculator & Visualizer

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATION LABORATORY

Euler's Method Engine

PreviousTaylor's Series Method

Unit 4•19 of 28

NextModified Euler's Method

Euler's Method First-Order Tangent Stepping

Step-by-Step Worked Example

Approximated Value: y(0.2) = 1.22000

Interactive Live Calculator & Visualizer

Euler's Method Engine

NETZ Engine

Euler's Method First-Order Tangent Stepping

Step-by-Step Worked Example

Approximated Value: y(0.2) = 1.22000

Interactive Live Calculator & Visualizer

Euler's Method Engine

Euler's Method
First-Order Tangent Stepping

Euler's Method
First-Order Tangent Stepping