Riccati equation, General - EqWorld - Mathematical...
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EqWorld http://eqworld.ipmnet.ru Exact Solutions > Ordinary Differential Equations > First-Order Ordinary Differential Equations > General Riccati Equation 23. y 0 x = f (x)y 2 + g(x)y + h(x). General Riccati equation. 1 ◦ . Given a particular solution y 0 = y 0 (x) of the Riccati equation, the general solution can be written as: y = y 0 (x) + Φ(x) • C - Z f (x)Φ(x) dx ‚ -1 , where Φ(x) = exp ‰ Z £ 2f (x)y 0 (x) + g(x) ∕ dx , C is an arbitrary constant. To the particular solution y 0 (x) there corresponds C = ∞. 2 ◦ . The substitution u(x) = exp h - Z f (x)y(x) dx i reduces the general Riccati equation to a second-order linear equation: f (x)u 00 xx - £ f 0 x (x) + f (x)g(x) ∕ u 0 x + f 2 (x)h(x)u =0, which often may be easier to solve than the original Riccati equation. 3 ◦ . See also special cases of the Riccati equation: • special Riccati equation , • Riccati equation, special case 1 , • Riccati equation, special case 2 , • Riccati equation, special case 3 , • Riccati equation, special case 4 , • Riccati equation, special case 5 , • Riccati equation, special case 6 , • Riccati equation, special case 7 , • Riccati equation, special case 8 , • Riccati equation, special case 9 , • Riccati equation, special case 10 , • Riccati equation, special case 11 , • Riccati equation, special case 12 , • Riccati equation, special case 13 , • Riccati equation, special case 14 , • Riccati equation, special case 15 , • Riccati equation, special case 16 . References Murphy, G. M., Ordinary Differential Equations and Their Solutions, D. Van Nostrand, New York, 1960. Reid, W. T., Riccati Differential Equations, Academic Press, New York, 1972. Kamke, E., Differentialgleichungen: L¨ osungsmethoden und L¨ osungen, I, Gew¨ ohnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition , Chapman & Hall/CRC, Boca Raton, 2003. General Riccati equation Copyright c 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
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Exact Solutions > Ordinary Differential Equations > First-Order Ordinary Differential Equations >General Riccati Equation
23. y′x = f (x)y2 + g(x)y + h(x).
General Riccati equation.
1. Given a particular solutiony0 = y0(x) of the Riccati equation, the general solution can be writtenas:
y = y0(x) + Φ(x)
[C −
∫f (x)Φ(x) dx
]−1
,
where
Φ(x) = exp
∫ [2f (x)y0(x) + g(x)
]dx
,
C is an arbitrary constant. To the particular solutiony0(x) there correspondsC = ∞.
2. The substitutionu(x) = exp
[−
∫f (x)y(x) dx
]
reduces the general Riccati equation to a second-order linear equation:
f (x)u′′xx −[f ′x(x) + f (x)g(x)
]u′x + f 2(x)h(x)u = 0,
which often may be easier to solve than the original Riccati equation.
3. See also special cases of the Riccati equation:• special Riccati equation,• Riccati equation, special case 1,• Riccati equation, special case 2,• Riccati equation, special case 3,• Riccati equation, special case 4,• Riccati equation, special case 5,• Riccati equation, special case 6,• Riccati equation, special case 7,• Riccati equation, special case 8,• Riccati equation, special case 9,• Riccati equation, special case 10,• Riccati equation, special case 11,• Riccati equation, special case 12,• Riccati equation, special case 13,• Riccati equation, special case 14,• Riccati equation, special case 15,• Riccati equation, special case 16.
ReferencesMurphy, G. M., Ordinary Differential Equations and Their Solutions, D. Van Nostrand, New York, 1960.Reid, W. T., Riccati Differential Equations, Academic Press, New York, 1972.Kamke, E., Differentialgleichungen: Losungsmethoden und Losungen, I, Gewohnliche Differentialgleichungen, B. G.
Teubner, Leipzig, 1977.Polyanin, A. D. and Zaitsev, V. F.,Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman
& Hall/CRC, Boca Raton, 2003.
General Riccati equation
Copyright c© 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
