Jacobian differential equation

The Jacobian differential equation named after Carl Gustav Jacob Jacobi is a nonlinear ordinary differential equation of the first order of form

${\displaystyle y'=f\left({\frac {ax+by+c}{\alpha x+\beta y+\gamma }}\right)\ .}$

An important special case is the Euler-homogeneous differential equation (after Leonhard Euler ), also called the similarity differential equation ,

${\displaystyle y'=f\left({\frac {y}{x}}\right).}$

Transformation to Euler-homogeneous differential equation

A case distinction must be made here according to whether it disappears or not. ${\displaystyle \det {\begin{pmatrix}a&b\\\alpha &\beta \\\end{pmatrix}}}$

Non-vanishing determinant

There are (unambiguous) paths with ${\displaystyle \det {\begin{pmatrix}a&b\\\alpha &\beta \\\end{pmatrix}}\neq 0}$ ${\displaystyle x^{\star },y^{\star }\in \mathbb {R} }$

${\displaystyle {\begin{array}{lcll}&ax^{\star }+by^{\star }&=&-c\\\wedge &\alpha x^{\star }+\beta y^{\star }&=&-\gamma \ .\\\end{array}}}$

Then follows

${\displaystyle {\frac {ax+by+c}{\alpha x+\beta y+\gamma }}={\frac {a(x-x^{\star })+b(y-y^{\star })}{\alpha (x-x^{\star })+\beta (y-y^{\star })}}={\frac {a+b{\frac {y-y^{\star }}{x-x^{\star }}}}{\alpha +\beta {\frac {y-y^{\star }}{x-x^{\star }}}}}\ .}$

The following applies: For every solution of the Euler homogeneous differential equation

${\displaystyle u'=f\left({\frac {a+b{\frac {u}{x}}}{\alpha +\beta {\frac {u}{x}}}}\right)=g\left({\frac {u}{x}}\right)\ ,\ g(s):=f\left({\frac {a+bs}{\alpha +\beta s}}\right)}$

is the solution of the original Jacobian differential equation, because one obtains ${\displaystyle y(x):=y^{\star }+u(x-x^{\star })}$

${\displaystyle y'(x)=u'(x-x^{\star })=f\left({\frac {a+b{\frac {u(x-x^{\star })}{x-x^{\star }}}}{\alpha +\beta {\frac {u(x-x^{\star })}{x-x^{\star }}}}}\right)=f\left({\frac {a+b{\frac {y(x)-y^{\star }}{x-x^{\star }}}}{\alpha +\beta {\frac {y(x)-y^{\star }}{x-x^{\star }}}}}\right)=f\left({\frac {ax+by(x)+c}{\alpha x+\beta y(x)+\gamma }}\right).}$

Thus, solving a Jacobian differential equation is reduced to solving an Euler-homogeneous differential equation.

Vanishing determinant

Be now . There are three cases to be distinguished. ${\displaystyle \det {\begin{pmatrix}a&b\\\alpha &\beta \\\end{pmatrix}}=0}$

• The case ${\displaystyle b=\beta =0}$

This case is trivial because the right side differential equation no longer depends on. ${\displaystyle y}$

• The case ${\displaystyle a=\lambda \alpha ,b=\lambda \beta ,\beta \neq 0}$

For all solutions of the separated differential equation ${\displaystyle z}$

${\displaystyle z'=\alpha +\beta f\left({\frac {\lambda z+c}{z+\gamma }}\right)}$

is the solution of the Jacobian differential equation, because it holds ${\displaystyle y(x):={\frac {1}{\beta }}(z(x)-\alpha x)}$

${\displaystyle y'(x)={\frac {1}{\beta }}(\alpha +\beta f\left({\frac {\lambda z(x)+c}{z(x)+\gamma }}\right)-\alpha )=f\left({\frac {\lambda (\alpha x+\beta y(x))+c}{\alpha x+\beta y(x)+\gamma }}\right)=f\left({\frac {ax+by(x)+c}{\alpha x+\beta y(x)+\gamma }}\right)\ .}$

So here the procedure of separating the changeable is applicable.

• The case ${\displaystyle \alpha =\beta =0,b\neq 0}$

This is analogous to the previous case: For all solutions of the separated differential equation ${\displaystyle z}$

${\displaystyle z'=a+bf\left({\frac {z+c}{\gamma }}\right)}$

is the solution of the Jacobian differential equation. ${\displaystyle y(x):={\frac {1}{b}}(z(x)-ax)}$

Transformation of the Euler homogeneous equation to the separation of the variables

An Euler-homogeneous differential equation is given . For every solution of the separated differential equation ${\displaystyle y'=g\left({\frac {y}{x}}\right)}$ ${\displaystyle z}$

${\displaystyle z'(x)={\frac {1}{x}}(g(z)-z)}$

is the solution of the Euler homogeneous differential equation ${\displaystyle y(x):=x\cdot z(x)}$

${\displaystyle y'(x)=z(x)+xz'(x)=g(z(x))=g\left({\frac {y(x)}{x}}\right)\ .}$

The differential equation for can be treated further with the method of separating the variables . ${\displaystyle z}$

literature

• Harro Heuser: Ordinary differential equations . 3. Edition. BG Teubner Stuttgart, 1995, ISBN 3-519-22227-2

Individual evidence

1. ^ Heidrun Günzel: Ordinary differential equations , Oldenbourg-Verlag, 2008, ISBN 978-3486-58555-1 , p. 55