Riccati equation

In mathematics, a Riccati equation is any ordinary differential equation that has the form

Related Topics:
Mathematics - Ordinary differential equation

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: y' = q_0(x) + q_1(x) , y + q_2(x) , y^2

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It is named after Count Jacopo Francesco Riccati (1676-1754).

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The Riccati equation is not amenable to elementary techniques in solving differential equations, except as follows. If one can find any solution y_1, the general solution is obtained as

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: y = y_1 + u

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Substituting

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: y_1 + u

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in the Riccati equation yields

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: y_1' + u' = q_0 + q_1 cdot (y_1 + u) + q_2 cdot (y_1 + u)^2,

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and since

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: y_1' = q_0 + q_1 , y_1 + q_2 , y_1^2

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: u' = q_1 , u + 2 , q_2 , y_1 , u + q_2 , u^2

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or

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: u' - (q_1 + 2 , q_2 , y_1) , u = q_2 , u^2,

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which is a Bernoulli equation. Unfortunately, one finds y_1 by guessing. The substitution that is needed to solve this Bernoulli equation is

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: z = u^{1-2} = rac{1}{u}

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Substituting

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: y = y_1 + rac{1}{z}

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directly into the Riccati equation yields the linear equation

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: z' + (q_1 + 2 , q_2 , y_1) , z = -q_2

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The general solution to the Riccati equation is then given by

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: y = y_1 + rac{1}{z}

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where z is the general solution to the aforementioned linear equation.

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