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

${\displaystyle \frac{dy}{dx}}=f\left({\displaystyle \frac{y}{x}}\right).$ (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  $\ln |x|=\int \frac{dt}{f(t)-t}+\ln C$, whence the general solution of the homogeneous differential equation (1) is in a parametric form

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

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

It’s a question of the equation

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

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

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

Accordingly, the given differential equation has the parametric solution

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

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