DELIB: Airy function
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DELIB       Airy function

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Matched equation
  1. Differential equation name: Airy function
  2. Representation: y''-xy=0
  3. Initial conditions/boundary conditions: y(0)=1/(32/3Γ(2/3)) and y'(0)=-1/(31/3Γ(1/3))
  4. DELIB library label: L111a12345ai
Click here if there are question marks (?) in the above (your browser does not support ISO-8859-1 characters).
Explicit information about matched equation
  1. Exact solution: y=Ai(x)
    where "Ai" is the Airy function.
Referenced information about matched equation
Type of information Location
General Mathworld   Digital math library   Encyclopedia of Special Functions Abramowitz and Stegun
Differential equation NIST Digital math library (section AI.2)
Graphs and Visualizations NIST Digital math library (section AI.3)
Maclaurin Series NIST Digital math library (section AI.4)
Integral Representations NIST Digital math library (section AI.5)
Relations to Other Functions NIST Digital math library (section AI.6)
Asymptotic Expansions NIST Digital math library (section AI.7)
Modulus and Phase NIST Digital math library (section AI.8)
Zeros NIST Digital math library (section AI.9)
Integrals NIST Digital math library (section AI.10)
Methods of Computation NIST Digital math library (section AI.18)
Tables NIST Digital math library (section AI.19)
Approximations NIST Digital math library (section AI.20)
Software NIST Digital math library (section AI.21)
Related equations
  1. Removing initial conditions and/or boundary conditions from matched equation
    1. y''-xy=0 Airy equation (Equation L111a12345)
  2. Changing initial conditions and/or boundary conditions in matched equation
    1. y''=yx with y(0)=1/(31/6Γ(2/3)) and y'(0)=31/6/Γ(2/3)   Bairy function (Equation L111a12345bi)
Numerical solutions
  1. If you have installed the freely available ODE toolkit then this link will bring up the package with the above equation entered.
Main features of matched equation
  1. Type: ODE
  2. Linearity: linear
  3. Order: 2
  4. Degree: 1
  5. Number of independent variables: 1
  6. Number of dependent variables: 1
  7. Functions appearing in equation: polynomial
  8. Has initial and/or boundary conditions: yes
  9. Number of terms: 2
  10. Features of one of the terms:
    1. Order: 2
    2. Number of factors: 1
    3. Special form: a derivative of y
  11. Features of one of the terms:
    1. Order: 1
    2. Number of factors: 2
    3. Functions appearing in term: polynomial
    4. Features of one of the factors:
      1. Order: 2
      2. Special form: a derivative of y
To see complete list of features, press To see similar equations, press
Computer representations
Language Representation
Latex $y''-xy=0$ with $y(0)=\frac{1}{3^{2/3}\Gamma(\frac{2}{3})}$ with $y'(0)=-\frac{1}{3^{1/3}\Gamma(\frac{1}{3})}$
Mathematica D[y[x],{x,2}]-x*y[x]=0   (with initial conditions)
MathML <math> <list> <reln><eq/> <apply><plus/> <apply><diff/> <bvar> <ci>x</ci> <degree> <cn type="integer">2</cn> </degree> </bvar> <ci>y</ci> </apply> <apply><times/> <ci>x</ci> <ci>y</ci> </apply> </apply> <cn type="integer">0</cn> </reln> </list> </math>   (with initial conditions)
References
  1. M. Abramowitz and I. A. Stegun (Eds.) (1964), Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series No. 55. U.S. Government Printing Office, Washington, DC.
  2. N. Bleistein and R. A. Handelsman (1975). Asymptotic Expansions of Integrals, Holt, Rinehart, and Winston, New York.
  3. F. W. J. Olver (1974). Asymptotics and Special Functions, Academic Press. Reprinted by AK Peters, Wellesley, 1997.
  4. N. M. Temme (1996). Special Functions. An Introduction to the Classical Functions of Mathematical Physics, John Wiley & Sons, New York.
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