Understanding Engineering Mathematics

Note that because of the multi-valued nature of the inverse tan function we have to restrict
the region on which the DE is defined in order get a unique solution.
The previous cases dealt with either

dy

dx

=f(x) or

dy

dx

=g(y)

It is also quite easy to treat equations of the form

dy

dx

=f(x)g(y)

These are calledvariables separablebecause the expression fordy/dxcan be separated
into a product of separate functions ofxandyalone.
In this case we literally move ally’s to one side andx’s to the other:

dy

g(y)

=f(x)dx

hence, integrating both sides with respect to their respective variables

∫

dy

g(y)

=

∫

f(x)dx+C

Of course the left-hand side is now an integral with respect toy, the right-hand side
with respect tox. Note that we only need one arbitrary constant. Again, our only real
problem arises in actually doing the integrations.

Problem 15.7

Solve the initial value problems

(i)

dy

dx

=xy y=1whenx= 0

(ii)

dy

dx

=e^2 xYy y=0whenx= 0

(i) For

dy

dx

=xywe separate to give

dy

y

=xdx

so ∫

dy

y

=

∫

xdx+C

and integrating both sides with respect to their respective variables gives

lny=

x^2

2

+C