PlanetMath: ODE types reductible to the variables separable case

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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

and

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 , , then also , represents an integral curve for any constant . 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

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

and

are given constants. If

is constant, then $\frac{dy}{dx}$ is constant, and we see that the lines

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

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

as a function of

. Using still (4) gives also

. 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

and

. The substitution

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

given by $x-y = \pm1$ . Separating the variables

and

, 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.
Accessed 370 times total.

Classification:

AMS MSC:	34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
	34A09 (Ordinary differential equations :: General theory :: Implicit equations, differential-algebraic equations)

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