DELIB: Airy equation
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DELIB       Airy equation

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Matched equation
  1. Differential equation name: Airy equation
  2. Representation: y''-xy=0
  3. DELIB library label: L111a12345
Explicit information about matched equation
  1. Exact solution: y=C1 Ai(x) + C2 Bi(x)
    where "Ai" is the Airy function and "Bi" is the Bairy function and the {Ci} are arbitrary constants.
Referenced information about matched equation
Type of information Location
General Mathworld   Digital math library   Abramowitz and Stegun
Notation NIST Digital math library (section AI.1)
Differential equation NIST Digital math library (section AI.2)
Graphs and Visualizations NIST Digital math library (section AI.3)
Related equations
  1. Modifying matched equation
    1. y''-xy-1/pi=0 Scorer functions (Equation 12350)
  2. Generalizing matched equation
    1. y''-xny=0 Equation 12347
    2. y''-a2f(x)y=0 Equation 12351
  3. Transformations of matched equation
    1. y'+y2-x=0 Equation 12351     (transformation in AI.2.14 in (NIST Digital math library (section AI.2))
    2. y'''-4xy'-2y=0 Equation 12350     (transformation in AI.11.1 in (NIST Digital math library (section AI.11))
  4. Adding initial conditions and/or boundary conditions to matched equation
    1. y''-yx=0 with y(0)=1/(32/3Γ(2/3)) and y'(0)=-1/(31/3Γ(1/3))   Airy function (Equation L111a12345ai)
    2. y''-yx=0 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 matched equation.
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: no
  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 $y''-xy=0$
Mathematica D[y[x],{x,2}]-x*y[x]=0
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>
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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