Mathematical Methods for Physics and Engineering : A Comprehensive Guide

14.2 FIRST-DEGREE FIRST-ORDER EQUATIONS

which may then be integrated directly to give

μ(x)y=

∫

μ(x)Q(x)dx. (14.16)

The required integrating factorμ(x) is determined by the first equality in (14.15),

i.e.
d
dx

(μy)=μ

dy

dx

+

dμ

dx

y=μ

dy

dx

+μP y,

which immediately gives the simple relation

dμ

dx

=μ(x)P(x) ⇒ μ(x)=exp

{∫

P(x)dx

}

. (14.17)

Solve

dy

dx

+2xy=4x.

The integrating factor is given immediately by

μ(x)=exp

{∫

2 xdx

}

=expx^2.

Multiplying through the ODE byμ(x)=expx^2 and integrating, we have

yexpx^2 =4

∫

xexpx^2 dx=2expx^2 +c.

The solution to the ODE is therefore given byy=2+cexp(−x^2 ).

Solution method.Rearrange the equation into the form (14.14) and multiply by the

integrating factorμ(x)given by (14.17). The left- and right-hand sides can then be

integrated directly, giving y from (14.16).

14.2.5 Homogeneous equations

Homogeneous equation are ODEs that may be written in the form

dy

dx

=

A(x, y)

B(x, y)

=F

(y

x

)

, (14.18)

whereA(x, y)andB(x, y) are homogeneous functions of the same degree. A

functionf(x, y) is homogeneous of degreenif, for anyλ,itobeys

f(λx, λy)=λnf(x, y).

For example, ifA=x^2 y−xy^2 andB=x^3 +y^3 then we see thatAandBare

both homogeneous functions of degree 3. In general, for functions of the form of

AandB, we see that for both to be homogeneous, and of the same degree, we

require the sum of the powers inxandyin each term ofAandBto be the same