The Chemistry Maths Book, Second Edition

320 Chapter 11First-order differential equations

(iii).

Separation of the variables and integration gives

and the general solution is

By the initial condition,y 1 = 1122 whenx 1 = 11 ,

so thatc 1 = 11 and the required particular solution is

0 Exercises 13–22

Reduction to separable form

Some first-order differential equations that are not separable can be made so by means

of a suitable change of variables. One such example is when the functionF(x,y)on

the right of the general form (11.8) has the property

F(λx, λy) 1 = 1 F(x,y) (11.14)

where λis an arbitrary number or function; that is, ‘scaling’ both xand yby the same

factor leaves the function unchanged. A function of xand yhas this property if it can

be expressed as a function ofy 2 x. For example,

Then

Fxy f (11.15)

y

x

f

y

x

()λλ Fxy()

λ

λ

,=











=











=,

Fxy

xy

xy

y

x

y

x

f

y

x

(),=

• 
  

=











+











=













22

1

yx

x

()=

• 
  

1

1

3

y

c

()1

1

2

1

1

==

• 
  

yx

xc

()=

• 
  

1

3

−= , − = , =+

dy

y

xdx

dy

y

xdx

y

xc

2

2

2

23

33

1

ZZ

dy

dx

+=, = 301 xy y

1

2

22

()