**Unformatted text preview: **20 1 Solutions , . Now integrate both sides to get
where we computed by parts two times. Dividing by gives .
9. In this case
Now multiply to get
to
twice. Dividing by and the integrating factor is
.
, which simpliﬁes
. Now integrate both sides to get
, where we computed
by parts
gives
. 10. In standard form this equation becomes Using partial fractions we get , an antiderivative is
, and the integrating factor is
.
Now multiply by the integrating factor to get the left hand side of which is a perfect derivative and taking antiderivatives of both sides gives
is a constant. Now multiply by
to get
general solution. . Thus where
for the 11. In standard form we get
. An integrating factor is
Thus
Integrating both sides gives
, where the integral of the right hand side is done
by parts. Now divide by the integrating factor
to get
.
12. The given diﬀerential equation is in standard form,
tiderivative is
, and the integrating factor is
multiply by the integrating factor to get the left hand side of which is a perfect derivative If
then taking antiderivatives of both sides gives
where
is a constant. Now multiply by
to get
the general solution. In the case
then
and , an an. Now . Thus for
. ...

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