Riccatic differential equation

Riccatic differential equations or Riccati differential equations are a special class of nonlinear ordinary first order differential equations . You own the form

${\displaystyle y'(x)=f(x)y^{2}(x)+g(x)y(x)+h(x)}$

with given functions , and . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle h}$

They are named after the mathematician Jacopo Francesco Riccati , an Italian count (1676–1754) who dealt intensively with the classification of differential equations and developed methods for reducing the order of equations.

A general solution of a Riccati differential equation is generally not possible, but one can be given if a specific solution is known.

Riccatic differential equations have the same name as two other types of equations that are important for various topics from applied mathematics to finance.

Transformation in the case of a known solution

Assuming that a solution has already been found (e.g. by guessing). Then the Riccatic differential equation can be completely solved, since finding the remaining solutions is now reduced to a Bernoullian differential equation , which can easily be solved. ${\displaystyle u}$

Formulation of the transformation theorem

There are as well a solution to the differential equation riccatischen ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle u\colon (a,b)\rightarrow \mathbb {R} }$

${\displaystyle \ u'(x)=f(x)u^{2}(x)+g(x)u(x)+h(x)}$

and a solution to Bernoulli's differential equation ${\displaystyle z}$

${\displaystyle z'(x)=f(x)z(x)^{2}+{\Big (}2u(x)f(x)+g(x){\Big )}z(x)\ .}$

Then

${\displaystyle \ y(x):=z(x)+u(x)}$

the solution of the Riccatian differential equation

${\displaystyle y'(x)=f(x)y^{2}(x)+g(x)y(x)+h(x)\ ,\ y(x_{0})=y_{0}:=z(x_{0})+u(x_{0})\ .}$

proof

It applies

${\displaystyle {\begin{array}{lll}y'(x)&=&z'(x)+u'(x)\\&=&f(x)z(x)^{2}+{\Big (}2u(x)f(x)+g(x){\Big )}z(x)+f(x)u^{2}(x)+g(x)u(x)+h(x)\\&=&f(x){\Big [}z(x)^{2}+2u(x)z(x)+u^{2}(x){\Big ]}+g(x){\Big [}z(x)+u(x){\Big ]}+h(x)\\&=&f(x)y^{2}(x)+g(x)y(x)+h(x)\ ,\\\end{array}}}$

while the initial value is trivially fulfilled.

Conversion to a linear differential equation of the second order

In general, regardless of whether a special solution has been found, the Riccatic differential equation can be transformed to a linear differential equation of the second order with non-constant coefficients. Should the coefficients happen to be constant, this transformed equation can easily be completely solved with the help of the characteristic equation . In the case of non-constant coefficients, the linear form of the Riccatian differential equation can also be very difficult to solve.

Formulation of the transformation theorem

Let it be as well as continuously differentiable and a solution of the linear differential equation of the second order ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle f\colon (a,b)\rightarrow \mathbb {R} \setminus \{0\}}$ ${\displaystyle z}$

${\displaystyle z''(x)-\left[g(x)+{\frac {f'(x)}{f(x)}}\right]\cdot z'(x)+{\Big [}f(x)h(x){\Big ]}\cdot z(x)=0}$

with for everyone . Then ${\displaystyle z(x)\neq 0}$ ${\displaystyle x\in (a,b)}$

${\displaystyle y(x):=-{\frac {z'(x)}{f(x)z(x)}}}$

the solution of the Riccatian differential equation

${\displaystyle y'(x)=f(x)y^{2}(x)+g(x)y(x)+h(x)\ ,\ y(x_{0})=y_{0}:=-{\frac {z'(x_{0})}{f(x_{0})z(x_{0})}}\ .}$

proof

For the sake of clarity, the arguments are not recorded. According to the quotient rule applies

${\displaystyle {\begin{aligned}y'&={\frac {-fzz''+z'(f'z+fz')}{f^{2}z^{2}}}={\frac {-fz{\Big [}{\Big (}g+{\frac {f'}{f}}{\Big )}z'-fhz{\Big ]}+z'(f'z+fz')}{f^{2}z^{2}}}\\&=f{\frac {z'^{2}}{f^{2}z^{2}}}-g{\frac {z'}{fz}}+h=fy^{2}+gy+h,\end{aligned}}}$

while the initial value is trivially fulfilled.

Transformation to a system of linear differential equations

In addition to the conversion of the Riccati differential equation to a linear differential equation of the second order, a transformation to a linear differential equation system is also possible. This opens up further possible solutions. For example, in the case of constant coefficients with the matrix exponential function, an analytical solution of the original Riccati differential equation is obtained.

The ordinary differential equation of the second order (cf. formulation of the transformation theorem )

${\displaystyle z''(x)-k_{1}\cdot z'(x)+k_{2}\cdot z(x)=0}$

can be traced back to a system of ordinary first-order differential equations with the substitution and . The linear homogeneous differential equation system follows ${\displaystyle t_{1}=z}$ ${\displaystyle t_{2}=z'}$

${\displaystyle T^{\prime }(x)=A(x)\,T(x)}$

with the coefficient matrix ${\displaystyle A(x)\in \mathbb {R} ^{2\times 2}}$

${\displaystyle A={\begin{pmatrix}0&1\\k_{2}(x)&k_{1}(x)\end{pmatrix}}}$

and the vector ${\displaystyle T(x)\in \mathbb {R} ^{2}}$

${\displaystyle T={\begin{pmatrix}t_{1}\\t_{2}\end{pmatrix}}.}$

If the coefficient matrix is ​​a continuous function of (i.e. also in the case of non-constant coefficients), then the associated initial value problem with the initial values always has a uniquely determined solution that is explained for all . In addition, the solution can be given in matrix notation using the matrix exponential function even in the case of non-constant coefficients. ${\displaystyle x}$ ${\displaystyle (x_{0},T_{0})\in \mathbb {R} ^{2+1}}$ ${\displaystyle x\in \mathbb {R} }$

literature

• Harro Heuser : Ordinary differential equations , Vieweg + Teubner 2009 (6th edition), ISBN 978-3-8348-0705-2
• Wolfgang Walter: Ordinary differential equations. 7th edition. Springer, Berlin / Heidelberg 2000, ISBN 3-540-67642-2

Individual evidence

1. ^ W. Walter: Ordinary differential equations. 2000, p. 94.
2. ^ W. Walter: Ordinary differential equations. 2000, p. 305.
3. ↑ T. Möller: Symbolic mathematics-based simulation of cylinder spaces for regenerative gas cycles. In: Int J Energy Environ Eng. Springer, Berlin / Heidelberg, Feb. 2015. (link.springer.com)
4. ↑ G. Merziger, Th. Wirth: Repetitorium der higher Mathematik. Binomi Verlag, Hannover 2006.
5. ^ S. Blanes, F. Casas, JA Oteo, J. Ros: The Magnus expansion and some of its applications. In: Physics Reports. Volume 470, Cornell University Library, 2009. (arxiv.org)