Eulers Method Systems
Description
eulers_method_2x2 implements Euler’s method for finding a numerical solution of the first-order system of ODEs,
(1)\begin{align} \frac{dx}{dt} & = f(t, x, y) \\ \frac{dy}{dt} & = g(t, x, y) \\ x(t_0) & = x_0 \\ y(t_0) & = y_0, \end{align}
The following Sage commands use Euler's method to generate a solution for
(2)\begin{align} \frac{dx}{dt} & = x + y + t \\ \frac{dy}{dt} & = x - y \\ x(0) & = 0 \\ y(0) & = 0, \end{align}
where $h = 1/3$ and $t$ ranges from $0$ to $1$. eulers_method_2x2 is primarily used for pedagogical purposes.
Sage Cell
Code
t, x, y = PolynomialRing(QQ,3,"txy").gens()
f = x+y+t
g = x-y
eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1)
Options
Option
We can also generate a list of points instead of a table.
Code
t, x, y = PolynomialRing(QQ,3,"txy").gens()
f = x+y+t
g = x-y
eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1, algorithm ="none")
Option
We can specify our solutions to be real instead of rational.
Code
RR = RealField(sci_not=0, prec=4, rnd='RNDU')
t, x, y=PolynomialRing(RR,3,"txy").gens()
f = x+y+t
g = x-y
eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1)
Tags
Primary Tags: Differential Equations
Secondary Tags: Numerical Methods: Systems; 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. eulers_method implements Euler’s method for finding a numerical solution of the first-order ODE $y′=f(x,y)$.
- 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
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Author: Sage Tutorial v8.1
Date: 13 Feb 2018 16:16
Submitted by: Tom Judson