Singular solution

From Academic Kids

A singular solution of a differential equation is a solution that satisfies the following conditions:

  1. It solves the original differential equation.
  2. It is tangent to every solution from the family of general solutions of the ODE. By tangent we mean that there is a point x where ys(x) = yc(x) and y's(x) = y'c(x) where yc is any general solution.

Usually, singular solutions appear in differential equations when there is a need to divide in a term that might be equal to zero. Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution.

Example

Consider the following Clairaut's equation:

<math> y(x) = x \cdot y' + (y')^2 \,\!<math>

We write y' = p and then

<math> y(x) = x \cdot p + (p)^2 \,\!<math>

Now, we shall take the differential according to x:

<math> p dx = dy = p ( dx ) + x ( dp ) + 2 p ( dp ) \,\!<math>

which by simple algebra yields

<math> 0 = ( 2 p + x )dp \,\!<math>

This condition is solved if 2p+x=0 or if dp=0.

If dp=0 it means that y' = p = c = Const and the general solution is:

<math> y_c(x) = c \cdot x + c^2 \,\!<math>

where c is determined by the initial value.

If x + 2p = 0 than we get that p = -(1/2)x and subsituting in the ODE gives

<math> y_s(x) = -(1/2)x^2 + (-(1/2)x)^2 = -(1/4) \cdot x^2 \,\!<math>

Now we shall check whether this a singular solution.

First condition of tangency: ys(x) = yc(x) . We solve

<math> c \cdot x + c^2 = y_c(x) = y_s(x) = -(1/4) \cdot x^2 \,\!<math>

to find the intersection point, which is (-2c, -c).

Second condition tangency: y's(x) = y'c(x) .
We calculate the derivatives:

<math> y_c'(-2 \cdot c) = c \,\!<math>
<math> y_s'(-2 \cdot c) = -(1/2) \cdot x |_{x = -2 \cdot c} = c \,\!<math>

We see that both requirements are satisfied and therefore ys is tangent to general solution yc. Hence,

<math> y_s(x) = -(1/4) \cdot x^2 \,\!<math>

is a singular solution for the family of general solutions

<math> y_c(x) = c \cdot x + c^2 \,\!<math>

of this Clairaut equation:

<math> y(x) = x \cdot y' + (y')^2 \,\!<math>

Note: The method shown here can be used as general algorithm to solve any Clairaut's equation, i.e. first order ODE of the form

<math> y(x) = x \cdot y' + f(y'). \,\!<math>

See also: caustic (mathematics).

Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools