IncludeSingularSolutions

IncludeSingularSolutions

is an option for DSolve that specifies whether singular solutions should be returned along with the general solution for a nonlinear ordinary differential equation.

Details

  • Singular solutions are also known as envelope solutions or equilibrium solutions.
  • Singular solutions cannot be obtained by assigning finite numerical values to the arbitrary constants in the general solution for a nonlinear differential equation. Instead, they can be obtained by constructing the envelope of the family of curves represented by the general solution.
  • For example, if the general solution of a first-order ODE is given by the equation , where is an arbitrary constant, then the singular solutions can be obtained by solving the envelope equations and .
  • The following illustration shows the singular solution (envelope) for a nonlinear ODE whose general solution is a family of straight lines.
  • Singular solutions are closely related to physical phenomena such as caustics and wavefronts in optics that can be explained using envelope constructions.
  • Possible settings for IncludeSingularSolutions are:
  • Falsereturn only general solutions depending on constants C[i]
    Truereturn both general and singular solutions

Examples

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Basic Examples  (2)

By default, DSolve returns the general solution for this ODE:

Use IncludeSingularSolutions to compute singular solutions along with the general solution:

Visualize the general solution along with the envelope formed by this family:

Find the general solution of a logistic equation:

The singular solutions in this case are the "equilibrium" solutions y=0 and y=1:

Visualize the general solution along with the equilibrium solutions:

Properties & Relations  (1)

Obtain the general solution of a nonlinear ODE:

Construct the envelope for the family of curves defined by the general solution:

Obtain the same result using IncludeSingularSolutions:

Visualize the general solution along with the envelope formed by this family:

Wolfram Research (2022), IncludeSingularSolutions, Wolfram Language function, https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html.

Text

Wolfram Research (2022), IncludeSingularSolutions, Wolfram Language function, https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html.

CMS

Wolfram Language. 2022. "IncludeSingularSolutions." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html.

APA

Wolfram Language. (2022). IncludeSingularSolutions. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html

BibTeX

@misc{reference.wolfram_2024_includesingularsolutions, author="Wolfram Research", title="{IncludeSingularSolutions}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html}", note=[Accessed: 21-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_includesingularsolutions, organization={Wolfram Research}, title={IncludeSingularSolutions}, year={2022}, url={https://reference.wolfram.com/language/ref/IncludeSingularSolutions.html}, note=[Accessed: 21-November-2024 ]}