ODE types solvable by two quadratures

ODE types solvable by two quadratures

The

second order ordinary differential equation

d

2

⁢

y

d

⁢

x

2

=

f

⁢

(

x

,

y

,

d

⁢

y

d

⁢

x

)

(1)

may in certain special cases be solved by using two

quadratures

, sometimes also by

reduction

to a

first order

differential equation

(

http://planetmath.org/ODE

) and a quadrature.

If the right hand side of (1) contains at most one of the quantities

x

,

y

and

d

⁢

y

d

⁢

x

, the

general solution

solution is obtained by two quadratures.

•

The equation

d

2

⁢

y

d

⁢

x

2

=

f

⁢

(

x

)

(2)

is considered here (

http://planetmath.org/EquationYFx

).

•

The equation

d

2

⁢

y

d

⁢

x

2

=

f

⁢

(

y

)

(3)

has as

constant

solutions all real roots of the equation

f

⁢

(

y

)

=

0

.  The other solutions can be gotten from the

normal system

d

⁢

y

d

⁢

x

=

z

,

d

⁢

z

d

⁢

x

=

f

⁢

(

y

)

(4)

of (3).  Dividing the equations (4) we get now

d

⁢

z

d

⁢

y

=

f

⁢

(

y

)

z

.  By

separation of variables

and integration we may write

z

2

2

=

∫

f

⁢

(

y

)

⁢

𝑑

y

+

C

1

,

whence the first equation of (4) reads

d

⁢

y

d

⁢

x

=

2

⁢

∫

f

⁢

(

y

)

⁢

𝑑

y

+

C

1

.

here the variables and integrating give the

general integral

of (3) in the form

∫

d

⁢

y

2

⁢

∫

f

⁢

(

y

)

⁢

𝑑

y

+

C

1

=

x

+

C

2

.

(5)

The integration constant (

http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation

)

C

1

has an influence on the form of the integral curves, but

C

2

only

translates

them in the direction of the

x

-axis.

•

The equation

d

2

⁢

y

d

⁢

x

2

=

f

⁢

(

d

⁢

y

d

⁢

x

)

(6)

is

equivalent

(

http://planetmath.org/Equivalent3

) with the normal system

d

⁢

y

d

⁢

x

=

z

,

d

⁢

z

d

⁢

x

=

f

⁢

(

z

)

.

(7)

If the equation

f

⁢

(

z

)

=

0

has real roots

z

1

,

z

2

,

…

,  these

satisfy

the latter of the equations (7), and thus, according to the former of them, the differential equation (6) has the solutions

y

:=

z

1

⁢

x

+

C

1

,

y

:=

z

2

⁢

x

+

C

2

,

…

.

The other solutions of (6) are obtained by separating the variables and integrating:

x

=

∫

d

⁢

z

f

⁢

(

z

)

+

C

.

(8)

If this

antiderivative

is

expressible in closed form

and if then the equation (8) can be solved for

z

, we may write

z

=

d

⁢

y

d

⁢

x

=

g

⁢

(

x

-

C

)

.

Accordingly we have in this case the general solution of the ODE (6):

y

=

∫

g

⁢

(

x

-

C

)

⁢

𝑑

x

+

C

′

.

(9)

In other cases, we express also

y

as a

function

of

z

.  By the

chain rule

, the normal system (7) yields

d

⁢

y

d

⁢

z

=

d

⁢

y

d

⁢

x

⋅

d

⁢

x

d

⁢

z

=

z

f

⁢

(

z

)

,

whence

y

=

∫

z

⁢

d

⁢

z

f

⁢

(

z

)

+

C

′

.

Thus the general solution of (6) reads now in a parametric form as

x

=

∫

d

⁢

z

f

⁢

(

z

)

+

C

,

y

=

∫

z

⁢

d

⁢

z

f

⁢

(

z

)

+

C

′

.

(10)

The equations 10 show that a

translation

of any integral curve yields another integral curve.

Title

ODE types solvable by two quadratures

Canonical name

ODETypesSolvableByTwoQuadratures

Date of creation

2015-03-20 17:04:58

Last modified on

2015-03-20 17:04:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 34A05

Synonym

second order ODE types solvable by quadratures

Related topic

ODETypesReductibleToTheVariablesSeparableCase

Generated on Fri Feb 9 20:39:49 2018 by

LaTeXML