### About

# Explicit second order differential equation

The simulation calculates solutions of ordinary explicit second order differential equations.

**y´ ´= d ^{2}y/ dx = f( y, 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 of

*y.*

With a second slider you can define the initial value for the first derivative. It corresponds to the gradient at the initial ordinate, which is symbolized by an arrow.

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.

Activating **start** for 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 see an interpolated 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 create
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 calculation by slider

**step**and look how different resolutions influence the result.

*,*

The smaller window shows the phase space projections

**y ´ = y ´( y ) **

**y´´ = y´´ ( y ) **

The thick points are the last ones calculated.

The **phase space diagrams **very distinctively demonstrate
the different character of solutions:

*convergence, divergence, periodic oscillation, oscillating divergence, oscillating convergence.*It is independent of the initial condition.

**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.

Differential equations of order n are separated into n coupled first order differential equations by substitution, and are calculated accordingly. For a 2nd order equation this leads to

*y´´ = f( y, y´, x) ➔ y´ = dy/dx and
y´´ = f( y, y´, x)*

In this simulation with a **ComboBox** the formula is:

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

**In all experiments study the phase space diagrams too! **

**E 1:** Run **cosine** and try the points and the trace
option.

What do the phase space projections mean?

Try different step widths.

**E 2:** Go **back**, and chose new initial conditions. * Start*
creates the solution, which is different from the first one.

Try *points* an *trace* option.

**E 3: ** Create a set of solutions with identical initial value
for *y *and different ones for *y´. *What is the result
of different *y´* for the sine function?

**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 a set of curves with different initial values
for *y* and *y*´, including negative ones. Interpret the
results by analyzing the differential equation..

**E 6:** First choose **Exponential, **then **Exponential
Damping**. Observe the phase space diagrams. What is the difference?
Change initial values and compare again.

**E 7: **Choose **hyperbolic sine** with default initial
values *y = 1 y´= 1*.

Now choose **hyperbolic cosine** with default initial values *y = 1
y´= 0*.

Analyze the phase space diagrams.

**Remarks: ** For the *normal* exponential the gradient at *x
= 0* is equal to the initial value of *y* and cannot be zero for
a meaningful exponential. Gradient zero for finite *y* is
characteristic for the hyperbolic cosine *(e ^{x }+ e^{- x})/2*,
gradients > 0 with initial

*y = 0*for the hyperbolic sine

*(e*. Imagine both functions mirrored at the zero-ordinate for completeness.

^{x}- e^{- x})/2

**E 8: **Choose** slowing oscillation** and study how the
dependence on *x* influences the periods. Edit the formula such
that frequency increases and slowing decreases.

**E 9: **Choose** increasing oscillation **and edit formulas
correspondingly. Compare the effect of proportional and of reciprocal
dependencies on *x*. Try * nonlinear* dependencies.

**E 10:** Choose **damped oscillation.** Check if periods are
constant (when clicking at a point its coordinates are shown in the
lower left corner).

**E 11: ** Choose **increasing oscillation** and compare the
results to those of **damped oscillation.** Superimpose both curves
and check if periods are equal.

**E 12**: Draw conclusions as to **which consequences different
terms in the differential equation have**. With that in mind,
construct differential equations that will show specific characteristics.

this file was created by Dieter Roess in March 2009

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

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

### Translations

Code | Language | Translator | Run | |
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### 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 displays the graphs for a first-order and second-order differential equation, determined by the function selected in the combo box.

The graph on the left shows the original curve ** y = f(x)**. The graph on the right shows the graph of

**against**

*y'***(in blue) and the graph of**

*y***against**

*y''***(in red).**

*y*The graphs can be viewed either as a set of points or as a continuous trace, by selecting the respective option on the top of the simulation. Press the reset button to reset the simulation, and the play button to play it (*this may be intuitive for most users, but I find that this instruction is occasionally useful.*)

Research

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