ODE types reductible to the variables separable case

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

)

⁢

d

⁢

x

+

Y

⁢

(

x

,

y

)

⁢

d

⁢

y

=

0

,

where

X

and

Y

are two homogeneous functions of the same degree (

http://planetmath.org/HomogeneousFunction

).  Therefore, if the equation is written as

d

⁢

y

d

⁢

x

=

-

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

d

⁢

y

d

⁢

x

=

f

⁢

(

y

x

)

.

(1)

Accordingly, if this ratio is constant, then also

d

⁢

y

d

⁢

x

is constant; thus all lines

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

=

C

⁢

x

⁢

(

t

)

,

y

=

C

⁢

y

⁢

(

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

y

x

:=

t

;

y

=

t

⁢

x

;

d

⁢

y

d

⁢

x

=

t

+

x

⁢

d

⁢

t

d

⁢

x

.

The equation takes the form

t

+

x

⁢

d

⁢

t

d

⁢

x

=

f

⁢

(

t

)

(2)

which shows that any root (

http://planetmath.org/Equation

)

t

ν

of the equality

f

⁢

(

t

)

=

t

gives a

singular solution

y

=

t

ν

⁢

x

.
The variables in (2) may be :

d

⁢

x

x

=

d

⁢

t

f

⁢

(

t

)

-

t

Thus one obtains

ln

⁡

|

x

|

=

∫

d

⁢

t

f

⁢

(

t

)

-

t

+

ln

⁡

C

, whence the

general solution

of the homogeneous differential equation (1) is in a

parametric form

x

=

C

⁢

e

∫

d

⁢

t

f

⁢

(

t

)

-

t

,

y

=

C

⁢

t

⁢

e

∫

d

⁢

t

f

⁢

(

t

)

-

t

.

II.  Equation of the form

y

′

= f(ax+by+c)

It’s a question of the equation

d

⁢

y

d

⁢

x

=

f

⁢

(

a

⁢

x

+

b

⁢

y

+

c

)

,

(3)

where

a

,

b

and

c

are given constants.  If

a

⁢

x

+

b

⁢

y

is constant, then

d

⁢

y

d

⁢

x

is constant, and we see that the lines

a

⁢

x

+

b

⁢

y

=

constant  are isoclines of the intgral curves of (3).

Let

a

⁢

x

+

b

⁢

y

+

c

:=

u

(4)

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

d

⁢

u

d

⁢

x

=

a

+

b

⁢

f

⁢

(

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

=

∫

d

⁢

u

a

+

b

⁢

f

⁢

(

u

)

+

C

,

y

=

1

b

⁢

(

u

-

c

-

a

⁢

∫

d

⁢

u

a

+

b

⁢

f

⁢

(

u

)

-

a

⁢

C

)

.

Example.

In the nonlinear equation

d

⁢

y

d

⁢

x

=

(

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

d

⁢

u

d

⁢

x

=

1

-

u

2

,

where one can separate the variables.  Since the right hand side has the zeros

u

=

±

1

,  the given equation has the singular solutions

y

given by

x

-

y

=

±

1

.  Separating the variables

x

and

u

, one obtains

d

⁢

x

=

d

⁢

u

1

-

u

2

,

whence

x

=

∫

d

⁢

u

(

1

+

u

)

⁢

(

1

-

u

)

=

1

2

⁢

∫

(

1

1

+

u

+

1

1

-

u

)

⁢

𝑑

u

=

1

2

⁢

ln

⁡

|

1

+

u

1

-

u

|

+

C

.

Accordingly, the given

differential equation

has the parametric solution

x

=

ln

⁡

|

1

+

u

1

-

u

|

+

C

,

y

=

ln

⁡

|

1

+

u

1

-

u

|

-

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

Generated on Fri Feb 9 20:40:14 2018 by

LaTeXML