Euler's Method for Ordinary Differential Equations

Euler's Method is the simplest first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

It works by taking small linear step increments along the tangent slope computed at the current point to approximate the curve of the solution function over an interval.

Formula for Euler's Method:

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

Where:

$x_n$ is the current point value of $x$ .
$y_n$ is the current estimated solution value of $y$ .
$h$ is the step size increment.
$f(x_n, y_n)$ is the slope function $\frac{dy}{dx}$ evaluated at $(x_n, y_n)$ .

Example of Euler's Method

Solve Differential Equation

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

Find $y(0.2)$ with $h = 0.1$

Step 1: Identify Parameters

Initial point $x_0 = 0, y_0 = 1$ , step size $h = 0.1$ , slope function $f(x, y) = x + y$ .

Step 2: First Iteration ( $x_1 = 0.1$ )

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

y_1 = 1 + (0.1)(1) = 1.1

Step 3: Second Iteration ( $x_2 = 0.2$ )

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

y_2 = 1.1 + (0.1)(1.2) = 1.22

Conclusion

\boxed{y(0.2) \approx 1.22000}

Therefore, using Euler's Method with step size $h = 0.1$ , the approximate value of $y$ at $x = 0.2$ is 1.22000 (exact analytical value 1.24281).

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Unit 4•19 of 28

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Euler's Method for Ordinary Differential Equations

Euler's Method is the simplest first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

It works by taking small linear step increments along the tangent slope computed at the current point to approximate the curve of the solution function over an interval.

Formula for Euler's Method:

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

Where:

$x_n$ is the current point value of $x$ .
$y_n$ is the current estimated solution value of $y$ .
$h$ is the step size increment.
$f(x_n, y_n)$ is the slope function $\frac{dy}{dx}$ evaluated at $(x_n, y_n)$ .

Example of Euler's Method

Solve Differential Equation

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

Find $y(0.2)$ with $h = 0.1$

Step 1: Identify Parameters

Initial point $x_0 = 0, y_0 = 1$ , step size $h = 0.1$ , slope function $f(x, y) = x + y$ .

Step 2: First Iteration ( $x_1 = 0.1$ )

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

y_1 = 1 + (0.1)(1) = 1.1

Step 3: Second Iteration ( $x_2 = 0.2$ )

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

y_2 = 1.1 + (0.1)(1.2) = 1.22

Conclusion

\boxed{y(0.2) \approx 1.22000}

Therefore, using Euler's Method with step size $h = 0.1$ , the approximate value of $y$ at $x = 0.2$ is 1.22000 (exact analytical value 1.24281).

Interactive Euler's Method Calculator

Euler's Method ODE Solver

PreviousTaylor's Series Method

Unit 4•19 of 28

NextModified Euler's Method

Euler's Method for Ordinary Differential Equations

Example of Euler's Method

Step 1: Identify Parameters

Step 2: First Iteration (x1=0.1x_1 = 0.1x1​=0.1)

Step 3: Second Iteration (x2=0.2x_2 = 0.2x2​=0.2)

Conclusion

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NETZ Engine

Euler's Method for Ordinary Differential Equations

Example of Euler's Method

Step 1: Identify Parameters

Step 2: First Iteration (x1=0.1x_1 = 0.1x1​=0.1)

Step 3: Second Iteration (x2=0.2x_2 = 0.2x2​=0.2)

Conclusion

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Euler's Method ODE Solver

Step 2: First Iteration ( $x_1 = 0.1$ )

Step 3: Second Iteration ( $x_2 = 0.2$ )

Step 2: First Iteration ( $x_1 = 0.1$ )

Step 3: Second Iteration ( $x_2 = 0.2$ )