# ODE types reductible to the variables separable case

There are certain of non-linear ordinary differential equations^{} of first order (http://planetmath.org/ODE) which may by a suitable substitution be to a form where one can separate (http://planetmath.org/SeparationOfVariables) the variables.

I. So-called homogeneous differential equation

This means the equation of the form

$$X(x,y)dx+Y(x,y)dy=0,$$ |

where $X$ and $Y$ are two homogeneous functions of the same degree (http://planetmath.org/HomogeneousFunction). Therefore, if the equation is written as

$$\frac{dy}{dx}=-\frac{X(x,y)}{Y(x,y)},$$ |

its right hand side is a homogeneous function of degree 0, i.e. it depends only on the ratio $y:x$, and has thus the form

$\frac{dy}{dx}}=f\left({\displaystyle \frac{y}{x}}\right).$ | (1) |

Accordingly, if this ratio is constant, then also $\frac{dy}{dx}$ is constant; thus all lines $\frac{y}{x}=$ constant are isoclines of the family of the integral curves which intersect any such line isogonally.

We can infer as well, that if one integral curve is represented by $x=x(t)$, $y=y(t)$, then also $x=Cx(t)$, $y=Cy(t)$ an integral curve for any constant $C$. Hence the integral curves are homothetic^{} with respect to the origin; therefore some people call the equation (1) a similarity equation.

For generally solving the equation (1), make the substitution

$$\frac{y}{x}:=t;y=tx;\frac{dy}{dx}=t+x\frac{dt}{dx}.$$ |

The equation takes the form

$t+x{\displaystyle \frac{dt}{dx}}=f(t)$ | (2) |

which shows that any root (http://planetmath.org/Equation) ${t}_{\nu}$ of the equality $f(t)=t$ gives a singular solution $y={t}_{\nu}x$. The variables in (2) may be :

$$\frac{dx}{x}=\frac{dt}{f(t)-t}$$ |

Thus one obtains $\mathrm{ln}|x|=\int \frac{dt}{f(t)-t}+\mathrm{ln}C$, whence the general solution of the homogeneous differential equation (1) is in a parametric form

$$x=C{e}^{{\scriptscriptstyle \int {\scriptscriptstyle \frac{dt}{f(t)-t}}}},y=Ct{e}^{{\scriptscriptstyle \int {\scriptscriptstyle \frac{dt}{f(t)-t}}}}.$$ |

II. Equation of the form y${}^{\mathrm{\prime}}$= f(ax+by+c)

It’s a question of the equation

$\frac{dy}{dx}}=f(ax+by+c),$ | (3) |

where $a$, $b$ and $c$ are given constants. If $ax+by$ is constant, then $\frac{dy}{dx}$ is constant, and we see that the lines $ax+by=$ constant are isoclines of the intgral curves of (3).

Let

$ax+by+c:=u$ | (4) |

be a new variable. It changes the equation (3) to

$\frac{du}{dx}}=a+bf(u).$ | (5) |

Here, one can see that the real zeros $u$ of the right hand side yield lines (4) which are integral curves of (3), and thus we have singular solutions. Moreover, one can separate the variables in (5) and integrate, obtaining $x$ as a function of $u$. Using still (4) gives also $y$. The general solution is

$$x=\int \frac{du}{a+bf(u)}+C,y=\frac{1}{b}\left(u-c-a\int \frac{du}{a+bf(u)}-aC\right).$$ |

Example. In the nonlinear equation

$$\frac{dy}{dx}={(x-y)}^{2},$$ |

which is of the type II, one cannot separate the variables $x$ and $y$. The substitution $x-y:=u$ converts it to

$$\frac{du}{dx}=1-{u}^{2},$$ |

where one can separate the variables. Since the right hand side has the zeros $u=\pm 1$, the given equation has the singular solutions $y$ given by $x-y=\pm 1$. Separating the variables $x$ and $u$, one obtains

$$dx=\frac{du}{1-{u}^{2}},$$ |

whence

$$x=\int \frac{du}{(1+u)(1-u)}=\frac{1}{2}\int \left(\frac{1}{1+u}+\frac{1}{1-u}\right)\mathit{d}u=\frac{1}{2}\mathrm{ln}\left|\frac{1+u}{1-u}\right|+C.$$ |

Accordingly, the given differential equation has the parametric solution

$$x=\mathrm{ln}\sqrt{\left|\frac{1+u}{1-u}\right|}+C,y=\mathrm{ln}\sqrt{\left|\frac{1+u}{1-u}\right|}-u+C.$$ |

## References

- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title | ODE types reductible to the variables separable case |

Canonical name | ODETypesReductibleToTheVariablesSeparableCase |

Date of creation | 2013-03-22 18:06:36 |

Last modified on | 2013-03-22 18:06:36 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 34A09 |

Classification | msc 34A05 |

Related topic | SeparationOfVariables |

Related topic | ODETypesSolvableByTwoQuadratures |

Related topic | TheoryForSeparationOfVariables |

Defines | homogeneous differential equation |

Defines | similarity equation |