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[parent] ODE types reductible to the variables separable case (Topic)

There are certain types of non-linear ordinary differential equations of first order which may by a suitable substitution be reduced to a form where one can separate the variables.

I. So-called homogeneous differential equation

This means the equation of the form

$\displaystyle X(x,\,y)dx+Y(x,\,y)dy = 0,$
where $ X$ and $ Y$ are two homogeneous functions of the same degree. Therefore, if the equation is written as
$\displaystyle \frac{dy}{dx} = -\frac{X(x,\,y)}{Y(x,\,y)},$
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
$\displaystyle \frac{dy}{dx} = f\left(\frac{y}{x}\right).$ (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)$ represents 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

$\displaystyle \frac{y}{x} := t; \quad y = tx; \quad \frac{dy}{dx} = t+x\frac{dt}{dx}.$
The equation takes the form
$\displaystyle t+x\frac{dt}{dx} = f(t)$ (2)

which shows that any root $ t_\nu$ of the equality $ f(t) = t$ gives a singular solution $ y = t_\nu x$. The variables in (2) may be separated:
$\displaystyle \frac{dx}{x} = \frac{dt}{f(t)\!-\!t}$
Thus one obtains $ \ln{\vert x\vert} = \int\!\frac{dt}{f(t)\!-\!t}+ \ln{C}$, whence the general solution of the homogeneous differential equation (1) is in a parametric form
$\displaystyle x = Ce^{\int\!\frac{dt}{f(t)\!-\!t}}, \quad y = Cte^{\int\!\frac{dt}{f(t)\!-\!t}}.$

II. Equation of the form y$ '$= 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

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

Example. In the nonlinear equation

$\displaystyle \frac{dy}{dx} = (x-y)^2,$
which is of the type II, one cannot separate the variables $ x$ and $ y$. The substitution $ x-y := u$ converts it to
$\displaystyle \frac{du}{dx} = 1-u^2,$
where one can separate the variables. Since the right hand side has the zeros $ u = \pm1$, the given equation has the singular solutions $ y$ given by $ x-y = \pm1$. Separating the variables $ x$ and $ u$, one obtains
$\displaystyle dx = \frac{du}{1-u^2},$
whence
$\displaystyle x = \int\frac{du}{(1+u)(1-u)} = \frac{1}{2}\int\left(\frac{1}{1+u}+\frac{1}{1-u}\right)du = \frac{1}{2}\ln\left\vert\frac{1+u}{1-u}\right\vert+C.$
Accordingly, the given differential equation has the parametric solution
$\displaystyle x = \ln\sqrt{\left\vert\frac{1\!+\!u}{1\!-\!u}\right\vert}+C, \quad y = \ln\sqrt{\left\vert\frac{1\!+\!u}{1\!-\!u}\right\vert}-u\!+\!C.$

Bibliography

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



"ODE types reductible to the variables separable case" is owned by pahio.
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Also defines:  homogeneous differential equation, similarity equation

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Cross-references: function, real, curves, parametric form, general solution, singular solution, equality, origin, homothetic, intersect, integral curves, isoclines, lines, ratio, right hand side, homogeneous functions, equation, variables, ordinary differential equations
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This is version 7 of ODE types reductible to the variables separable case, born on 2008-06-04, modified 2008-06-12.
Object id is 10655, canonical name is ODETypesReductibleToTheVariablesSeparableCase.
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Classification:
AMS MSC34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
 34A09 (Ordinary differential equations :: General theory :: Implicit equations, differential-algebraic equations)

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