# 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:
•  False return only general solutions depending on constants C[i] True return 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: