### About

# First order differential equation

The simulation calculates solutions of ordinary explicit first order differential equations

**y´ = dy/dx = f(y,x)**

using the Runge−Kutta algorithm. In the left coordinate system the
abscissa represents *x*, the ordinate *y*.

When opening the file you see a fat red point at *x = 0*,
representing the** initial value** *y _{0}* at its
abscissa

*x*. You can change the initial value with the slider, more exactly and unlimited by typing a value into the number field. Two additional number fields are used to define

_{0}*x*and

_{0}*x*

_{max}_{ .}Default values are:

*y*You can also

_{0}= 1; x_{0}= 0; x_{max}= 3.*draw*the red point to create new initial conditions.

In the **comboBox **you can chose between a number of predefined
types of functions. Their formula is shown in field * y´ = ..*
. There you can edit formulas or input any arbitrary first order
explicit differential equation.

When activating **start** with the default equation, the differential
equation of the exponential function * y´= y *is
evaluated. Calculation stops as

*x = x*. At first you see a set of calculated points. You can choose the option

_{max}**trace**to get an interpolated smooth curve.

**Stop** stops the calculation; **back** leaves already calculated
points and sets back to the initial conditions. Changing these now
creates an additional curve at **start**. This way you can generate
sets of solutions for different initial conditions (the *Trace option*
would create jumps, which are avoided in the *points option*). **Clear**
resets and clears traces, but leaves parameters unchanged. **Reset **leads
back to default values.

After **back **you can change the

*x*-resolution of the calculation by slider

**step**and look if and how different resolution influence the result.

*,*

The smaller window shows the phase space projection

**y ´ = y´ ( y )**

The green point is the one last calculated.

The **phase space diagram **very distinctively demonstrates
the different character of solutions:

*convergence, divergence, periodic oscillation, oscillating divergence, oscillating convergence.*It is independent of the initial condition. The order of predefined functions follows these characteristics.

**Numerical integration of differential equations with EJS**

Using EJS it is very easy to solve differential equations. Several
algorithms for different methods are programmed and can be chosen at page*
Evolution* with a mouse click. The steps of the variable *x* are
automatically calculated, when the difference *dx* has been
defined. For a first order differential equation one defines:

*dy/dt = y´ = f( y, x ) *

In this simulation with a ComboBox the formula is:

*y´ *= (formula in field *y´*, evaluated for *x, *and
*y*)

In February 2011 **EJS 4.3** presents the following methods:

- Euler
- Euler−Richardson
- Velocity Verlet
- Runge− Kutta, 4- steps
- Bogacki−Shampine 3(2)
- Cah−Karp 5(4)
- Fehlberg 8(7)
- Dormand−Prince 5(4)
- Dormand−Prince 8(5
- Radau 5(4)
- OSS3

**E 1:** Run **exponential function** and try the points and
the trace option. After **back** try different step widths (**speed ).**
With the exponential you will see no difference within the given range
of the slider.

**E 2:** Go **back**, and chose a new initial condition. **Start**
creates the solution, which is different from the first one. Try the
points and trace options.

**E 3: ** Create a set of solutions with identical *x _{0}*
and different

*y*. Explain what you see.

_{0}

**E 4: **Create a set of solutions with identical *y _{0}*
and different

*x*. Why do you see curves that are shifted parallel?

_{0}

**E 5: **Create sets of solutions including negative initial
values *y _{0}*. Discuss the result as consequence of the
differential equation.

**E 6:** Choose **exponential damping**. Compare solutions
for positive and negative initial values to those of the exponential.

**E 7: **Choose any of the **oscillating** **function s**
and try to guess the trace before calculation by studying the
differential equation.

**E 8: **Edit the exponential by adding factors *g(x)*,
especially periodic ones. Try to understand what you observe.

**E 9: **Run the exponential once. Choose **back .** Edit
the formula by setting

*e^x*instead of

*y.*In a second run you will see the same result: The algorithm now performs ordinary

*(numerical calculation of the*

**Integration***. This works for the exponential, because*

**antiderivative =**indefinite integral**)***y*is not explicit in

*dy/dx = e^x.*

*.*

**E 10:** Choose any other function not containing *y*, e.g. *sin(x)*.
For this you get *-cos(x)* as indefinite integral,with the initial
value as * integration constant. *Compare the phase space
diagram of

*Sine*with that of the function type

*Periodical.*

**E11: ** Choose **constant velocity** and create a curve set
for different factors *y*, including zero. This again is a simple
integration. Edit the differential equation in such a way that an
initial starting location (x_{0}) is considered.

**E12: **Choose **constant acceleration** and create a curve set
for different factors of *y*, including zero. This again is a
simple integration. Edit the differential equation in such a way that an
initial velocity is considered.

**E 13: ** Extend the differential equation by more complicated
functions of *y* and let yourself be surprised by what happens. Try
to define and understand interrelated families of differential
equations. Are there more classes than *convergent, divergent, periodic*?

**E14:** After **back** change the step width for different
functions and observe the result. Which characteristics define the
maximum width below which no deviations are observed?

This file was created by Dieter Roess in March 2009

This simulation is part

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

This simulation provides a visualisation of first-order ordinary differential equations, available from a selection in the combo box provided. A total of 9 different types of equations can be selected and graphed on the plotting panel in the simulation.

The display of the plot can be toggled between **Points** (discrete points) and **Trace** (a continuous curve) in the combo box to the right of the formula box.

The initial starting point of y can be toggled with the slider labelled y0.

There are 2 plotting panels. The one on the left represents the graph of ** y** against

*x*, while the one on the right represents the graph of the derivative

**against**

*y'***.**

*y*The available functions along with their associated differential equations are as follows:

- Exponential (** y = e^x**):

*y' = y*- Exponential Damping (** y = e^(-x)**):

*y' = -y*- Periodic: *y' = ysin(x) *

- Oscillation Divergent: *y' = y(x/5)sin(x)*

- Oscillation Convergent: *y' = (y/x)sin(x)*

- Oscillation Quadratic: *y' = ysin((x-10)^2)*

- Sine Exponent: *y' = y^sin(x)*

- Constant Velocity: *y' = 1*

- Constant Acceleration: *y' = x*

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