Difference between revisions of "Riccati equation"
From timescalewiki
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| − | Let $\mathbb{T}$ be a [[time scale]]. The Riccati equation is the nonlinear [[dynamic equation]] defined by | + | Let $\mathbb{T}$ be a [[time scale]]. The [[self-adjoint]] equation is $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. The Riccati equation is the nonlinear [[dynamic equation]] defined by |
$$z^{\Delta}(t) + q(t) + \dfrac{z^2(t)}{p(t)+\mu(t)z(t)}=0,$$ | $$z^{\Delta}(t) + q(t) + \dfrac{z^2(t)}{p(t)+\mu(t)z(t)}=0,$$ | ||
where $p(t)+\mu(t)z(t)>0$ for all $t \in \mathbb{T}^{\kappa}$. | where $p(t)+\mu(t)z(t)>0$ for all $t \in \mathbb{T}^{\kappa}$. | ||
| Line 5: | Line 5: | ||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| − | <strong>Theorem:</strong> | + | <strong>Theorem:</strong> There exists a solution $y$ of the self-adjoint equation with $y(t) \neq 0$ for all $t \in \mathbb{T}$ if and only if the Riccati equation has a solution $z$ related by $z=\dfrac{py^{\Delta}}{y}$. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
Revision as of 22:45, 27 June 2015
Let $\mathbb{T}$ be a time scale. The self-adjoint equation is $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. The Riccati equation is the nonlinear dynamic equation defined by $$z^{\Delta}(t) + q(t) + \dfrac{z^2(t)}{p(t)+\mu(t)z(t)}=0,$$ where $p(t)+\mu(t)z(t)>0$ for all $t \in \mathbb{T}^{\kappa}$.
Properties
Theorem: There exists a solution $y$ of the self-adjoint equation with $y(t) \neq 0$ for all $t \in \mathbb{T}$ if and only if the Riccati equation has a solution $z$ related by $z=\dfrac{py^{\Delta}}{y}$.
Proof: █
