Math 2.0

Ordinary differential equations Definition

Riccati equation

Builds on Ordinary differential equationLinear first-order equationBernoulli equation

Definition

Let II be an open interval and let q0,q1,q2:I→ℝq_0, q_1, q_2 : I \to \mathbb{R} be continuous functions, with q0q_0 and q2q_2 not identically zero. A Riccati equation is an ordinary differential equation of the form y′=q0(x)+q1(x) y+q2(x) y2.y' = q_0(x) + q_1(x)\, y + q_2(x)\, y^2.

Remarks

If q2=0q_2 = 0, the equation is linear. If q0=0q_0 = 0, it is a Bernoulli equation with n=2n = 2. If y1y_1 is one solution, then the substitution y=y1+1/vy = y_1 + 1/v gives a linear first-order equation for vv. If q2q_2 has no zero and is differentiable, the substitution y=−u′/(q2u)y = -u' / (q_2 u) gives the second-order linear equation u′′−(q1+q2′/q2)u′+q0q2 u=0u'' - \left(q_1 + q_2'/q_2\right) u' + q_0 q_2\, u = 0. Example: y′=1+y2y' = 1 + y^2 has the solutions y=tan⁡(x+C)y = \tan(x + C).

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  1. Differential equation concept
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  5. Linear first-order equation definition
  6. Bernoulli equation definition
  7. Riccati equation definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Riccati_equation, lead section
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