Understanding Engineering Mathematics

(やまだぃちぅ) #1
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.

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