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Duffing equation

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1: 30.2 Differential Equations
§30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
2: 31.2 Differential Equations
§31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
3: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
4: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
15.10.1 z ( 1 - z ) d 2 w d z 2 + ( c - ( a + b + 1 ) z ) d w d z - a b w = 0 .
This is the hypergeometric differential equation. …
5: 32.2 Differential Equations
§32.2 Differential Equations
§32.2(i) Introduction
The six Painlevé equations P I P VI  are as follows: …
§32.2(ii) Renormalizations
6: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
7: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 - 1 ) w ′′ + ζ w + ( 4 q ζ 2 - 2 q - a ) w = 0 , ζ = cosh z .
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ζ 1 / 2 e ± 2 i h ζ as ζ in the respective sectors | ph ( i ζ ) | 3 2 π - δ , δ being an arbitrary small positive constant. …
8: 22.19 Physical Applications
Classical motion in one dimension is described by Newton’s equation
Case I: V ( x ) = 1 2 x 2 + 1 4 β x 4
This is an example of Duffing’s equation; see Ablowitz and Clarkson (1991, pp. 150–152) and Lawden (1989, pp. 117–119). …
§22.19(iii) Nonlinear ODEs and PDEs
9: 28.18 Integrals and Integral Equations
§28.18 Integrals and Integral Equations
10: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).