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Chapter 14: Secondorder ordinary differential equations
- Linear equations
- Representation of the general solution through a particular solution
- Equations containing power functions
- Equations containing exponential functions
- Equations containing hyperbolic functions
- Equations containing logarithmic functions
- Equations containing trigonometric functions
- Equations containing inverse trigonometric functions
- Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions
- Equations with arbitrary functions
- 14.1.10 Some Transformations
- Autonomous Equations y x x ′ ′ = F ( y , y x ′ ) $ y_{xx}^{''} = F(y, y_{x}^{'} ) $
- 14.2.1 Equations of the Form y x x ′ ′ - y x ′ = f ( y ) $ y_{xx}^{''} - y_{x}^{'} = f(y) $
- Equations of the Form y x x ′ ′ + f ( y ) y x ′ + y = 0 $ y_{xx}^{''} + f(y)y_{x}^{'} + y = 0 $
- 14.2.3 Lienard Equations y x x ′ ′ + f ( y ) y x ′ + g ( y ) = 0 $ y_{xx}^{''} + f(y)y_{x}^{'} + g(y) = 0 $
- Rayleigh Equations y x x ′ ′ + f ( y x ′ ) + g ( y ) = 0 $ y_{xx}^{''} + f(y_{x}^{'} ) + g(y) = 0 $
- Emden–Fowler Equation y x x ′ ′ = A x n y m $ y_{xx}^{''} = Ax^{n} y^{m} $
- Equations of the Form y x x ′ ′ = A 1 x n 1 y m 1 + A 2 x n 2 y m 2 $ y_{xx}^{''} = A_{1} x^{{n_{1} }} y^{{m_{1} }} + A_{2} x^{{n_{2} }} y^{{m_{2} }} $
- Generalized Emden-Fowler Equation
- Equations of the Form
- Other Equations (l 1 ≠ l 2)
- Equations of the Form y x x ″ = f ( x ) g ( y ) h ( y x ′ ) $ y_{xx}^{\prime\prime} = f(x)g(y)h(y_{x}^{\prime} ) $
- Equations of the Form y x x ″ = f ( x ) f ( y ) $ y_{xx}^{\prime\prime} = f(x)f(y) $
- Equations Containing Power Functions ( h ? $ (h\not \equiv $ const)
- Equations Containing Exponential Functions ( h ? $ (h\not \equiv $ const)
- Equations Containing Hyperbolic Functions ( h ? $ (h\not \equiv $ const)
- Equations Containing Trigonometric Functions ( h ? $ (h\not \equiv $ const)
- Some Transformations
- Some Nonlinear Equations with Arbitrary Parameters
- Equations Containing Power Functions
- Equations Containing Logarithmic Functions
- Equations Containing Trigonometric Functions
- Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions
- Equations Containing Arbitrary Functions
- Equations of the Form F ( x , y ) y x x ′ ′ + G ( x , y ) = 0 $ F(x, y)y_{xx}^{''} + G(x, y) = 0 $
- Equations of the Form F ( x , y ) y x x ′ ′ + G ( x , y ) y x ′ + H ( x , y ) = 0 $ F(x, y)y_{xx}^{''} + G(x, y)y_{x}^{'} + H(x, y) = 0 $
- Equations of the Form
- Equations Not Solved for Second Derivative
- Equations of General Form
- Equations Defined Parametrically
- Some Transformations
Abstract
A homogeneous linear equation of the second order has the general form f 2 ( x ) y x x ′ ′ + f 1 ( x ) y x ′ + f 0 ( x ) y = 0 . $$ {f_2}(x)y_{xx}^{''} + {f_1}(x)y_{x}^{'} + {f_0}(x)y = 0. $$
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