Understanding Engineering Mathematics

Dividing byxto make the coefficient of the derivative unity gives

dy

dx

+

1

x

y=x

Problem 15.10

Write down the DE satisfied by the integrating factor.

Multiplying through by an, as yet unknown, integrating factor,I,gives

I

dy

dx

+

I

x

y=xI

Now
d(Iy)
dx

=I

dy

dx

+

I

x

y

will be satisfied if

dI

dx

=

I

x

which is the required equation for the integrating factor. Note that it is separable.

Problem 15.11

Determine the integrating factorI.

Separating the variables of the last equation we have

dI

I

=

dx

x

or, integrating through
∫
dI
I

=

∫

dx

x

Performing the integrations gives

lnI=lnx

So in this case we have

I=x

for the integrating factor. Note that we don’t need to include an arbitrary constant at this
stage.

Problem 15.12

Solve the original equation.

Multiplying the linear form by the integrating factorxwe now have

x

dy

dx

+y=x^2