## Description

`eulers_method` implements Euler’s method for finding a numerical solution of the first-order ODE $y′=f(x,y)$. The $x$ column of the table increments from $x_0$ to $x_1$ by $h$ (so $(x_1 − x_0)/h$ must be an integer). In the $y$ column, the new $y$-value equals the old $y$-value plus the corresponding entry in the last column. Euler's method is used primarily for pedagogical purposes. The following Sage commands use Euler's method to generate a solution for

where $h = 1/2$ and $x$ ranges from $0$ to $1$.

## Sage Cell

#### Code

```
x,y = PolynomialRing(QQ,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1)
```

## Options

#### Option

To generate a list of points instead of a table, we can use the following commands.

#### Code

```
x,y = PolynomialRing(QQ,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1,algorithm="none")
```

#### Option

We can also generate solutions over the reals.

#### Code

```
RR = RealField(sci_not=0, prec=4, rnd='RNDU')
x,y=PolynomialRing(RR,2,"xy").gens()
eulers_method(5*x+y-5,0,1,1/2,1)
```

## Tags

Primary Tags: differential equations

Secondary Tags: numerical methods, euler

A list of possible tags can be found at The WeBWorK Open Problem Library. For linear algebra tags see the Curated Courses Project.

## Related Cells

- desolve. Solving ordinary differential equations with
`desolve`. - desolve_odeint. Solving ordinary differential equations numerically with
`desolve_odeint`. - Euler's Method for Systems.
`eulers_method_2x2`implements Euler’s method for finding a numerical solution of a $2 \times 2$ system of first-order ODEs. - desolve_laplace. Solving ordinary differential equations using Laplace transforms.
- Interact to plot direction fields and solutions for first order differential equations. A Sage interact for plotting direction fields for differential equations.

## Attribute

Permalink:

Author: Sage Tutorial v8.1

Date: 13 Feb 2018 15:51

Submitted by: Tom Judson