**Related Resources: math**

### Explicit Second Order Calculus Differential Equation Simulator Tool

**Explicit Second Order Calculus Differential Equation Simulator Tool**

Open (new Browser Window):

**Explicit second order differential equation Simulator Tool **

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 simulator**

Using this differential equation simulator 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´*)

This simulator 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*and different

_{0}*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.

Related:

- Differential Equations Kenneth A Stroud, Dexter J Booth
- Applied Dynamics in Engineering
- Mechanics and Machine Design, Equations and Calculators

Credits:

Created by Dieter Roess in March 2009

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