THE INTEGRATING FACTOR TECHNIQUE
for
First Order, Linear Differential Equations
Step 1. Write the differential equation in the form
dy
dx
+f (x)y = g(x) (or
+f (t)x = g(t))
dx
dt
Example: Write
dy
dy
= 3xy − 4 as
− 3xy = −4 (so here f (x) = −3x and g(x) = −4)
dx
dx
Example: Write
dx
dx 2
= −t2 x + sin(t) as
+ t x = sin(t) (so here f (t) = t2 and g(t) = sin(t))
dt
dt
Step 2. Find the antiderivative of f (x) and call it F (x). You do not need the “ + C”.
Step 3. The integration factor for this differential equation is formed by raising e to
the function F (x), that is, eF (x) . Multiply both sides of the DE (in the proper form) by the
integrating factor, eF (x) .
eF (x) (
dy
+ f (x)) = g(x)eF (x)
dx
eF (x)
or
dy
+ f(x)eF (x) = g(x)eF (x)
dx
Step 4. If you notice that the left hand side is the derivative of the product yeF (x) , then
the above can be written as
yeF (x) = g(x)eF (x)
Step 5. Integrate both sides of this new equation with respect to x
yeF (x) dx =
g(x)eF (x) dx
and notice that the left hand side is simply yeF (x) .
F (x)
Final Step. Now solve for y. From Step 5, we have ye
= g(x)eF (x) dx. After Þnding
the antiderivative of the right hand side and remembering to attach “+C”, divide both sides
by eF (x) .
EXAMPLES
1) Use the Integrating Factor Technique to solve
dy
dy
= −3y + 4 or in standard form
+ 3y = 4
dx
dx
so f (x) = 3, F (x) = 3x and the integrating factor is e3x . Now multiply both sides of the
standard form by the integrating factor.
dy
+ 3y = 4e3x
dx
3x
e
and notice that the left hand side can be written as a derivative.
e3x y
= 4e3x
Now integrate both sides with respect to x and get
4
e3x y = e3x + C
3
So
4
+ Ce−3x
3
which is be checked in this example by the separation of variables technique.
y=
2) The second example is more difficult in the sense that the differential equation cannot be
solved using separation of variables. So again, put in standard form and Þnd the integrating
factor.
dx
x
dx x
= − + 4 or in standard form
+ =4
dt
t
dt
t
1
so f (t) = , F (t) = ln(t) (we’ll assume that t > 0), and the integrating factor is eln(t) = t.
t
After multiplying both sides by the t, we have
dx x
t
+
dt
t
= 4t
or
(tx) = 4t.
Integrating with respect to t gives
tx = 2t2 + C
and solving for x
x = 2t +
C
.
t