Mathematical Methods for Physics and Engineering : A Comprehensive Guide

14.2 FIRST-DEGREE FIRST-ORDER EQUATIONS

are several different types of first-degree first-order ODEs that are of interest in

the physical sciences. These equations and their respective solutions are discussed

below.

14.2.1 Separable-variable equations

A separable-variable equation is one which may be written in the conventional

form

dy

dx

=f(x)g(y), (14.3)

wheref(x)andg(y) are functions ofxandyrespectively, including cases in

whichf(x)org(y) is simply a constant. Rearranging this equation so that the

terms depending onxand onyappear on opposite sides (i.e. are separated), and

integrating, we obtain
∫
dy
g(y)

=

∫

f(x)dx.

Finding the solutiony(x) that satisfies (14.3) then depends only on the ease with

which the integrals in the above equation can be evaluated. It is also worth

noting that ODEs that at first sight do not appear to be of the form (14.3) can

sometimes be made separable by an appropriate factorisation.

Solve

dy

dx

=x+xy.

Since the RHS of this equation can be factorised to givex(1 +y), the equation becomes
separable and we obtain
∫
dy
1+y

=

∫

xdx.

Now integrating both sides separately, we find

ln(1 +y)=

x^2

2

+c,

and so

1+y=exp

(

x^2

2

+c

)

=Aexp

(

x^2

2

)

,

wherecand henceAis an arbitrary constant.

Solution method.Factorise the equation so that it becomes separable. Rearrange

it so that the terms depending onxand those depending onyappear on opposite

sides and then integrate directly. Remember the constant of integration, which can

be evaluated if further information is given.