Understanding Engineering Mathematics

or sincee^0 =1,C=−^32 and the solution can be written

e−y=

3

2

−

e^2 x

2

Note that sincee−ymust be positive, we are here restricted toe^2 x≤3.

Solving forygives

−y=ln

∣∣

∣

∣

3

2

−

e^2 x

2

∣∣

∣

∣

or

y=−ln

∣

∣

∣

∣

2

3

−

e^2 x

2

∣

∣

∣

∣

The form of variables separable equations is rather restrictive. Even such a simple
function asF(x,y)=x+ywouldn’t fit into it. However, there are many types of equation
that may be reduced to variables separable by some kind of substitution. Consider, for
example the equation

dy

dx

=

x+y

x

=F(x,y)

whereF(x,y)is of the form

F(x,y)=f

(y

x

)

Such an equation is said to behomogeneous(notto be confused with later use of this

term). If we change our variables fromx,ytoxandv=

y

x

we havey=xvand so

dy

dx

=v+x

dv

dx

and the equation becomes

dy

dx

=v+x

dv

dx

=f(v)

or

x

dv

dx

=f(v)−v

This isseparableand its solution is

∫

dv

f(v)−v

=

∫

dx

x

+C=lnx+C

After evaluating the integral on the left we can then replacevbyy/xto get the solution
in terms ofxandy.