1. Method of Integrating Factors
One of the most fundamental methods for solving linear, first-order, ordinary differential equations is using
the integrating factor method. We’ll first start off by introducing the standard form of the class of differential
equations that are solvable using this method.
Definition A linear, first-order, ordinary differential equation in the dependent variable y is an
equation of the form
dy
dt
+ p(t)y = g(t). (1)
More generally, we often encounter an equation such as
a0(t)
dy
dt
+ a1(t)y = h(t).
However, we can put it into standard form (1), by simply dividing through by a0(t).
The goal of the integrating factor method is to make the left-hand-side an easily integrable expression by
multiplying by a function µ(t). This function µ(t) is our integrating factor, and we can define it as
µ(t) = e p(t)dt
. (2)
Then, if we multiply the differential equation (1) by the integrating factor µ(t), we have
e p(t)dt dy
dt
+ e p(t)dt
p(t)y = e p(t)dt
g(t).
Recall now from calculus that the derivative of an exponential function ef(x)
is f (x)ef(x)
. So if µ(t) is
defined as above in Eq. (2), then the derivative of µ(t) is
µ (t) = p(t)e p(t)dt
.
In particular, the left-hand side of our differential equation multiplied by the integrating factor is
e p(t)dt dy
dt
+ e p(t)dt
p(t)y =
dy
dt
µ(t) + y
dµ
dt
.
Recall again from calculus that the derivative of an expression of the form f(x)g(x) is f (x)g(x)+f(x)g (x).
Using this fact, we can see that the left-hand-side of our equation is the derivative of the product of y(t) and
µ(t). With all of this, our differential equation now looks like
y(t)e p(t)dt
= e p(t)dt
g(t), (3)
We are now ready to find a solution. The last steps we have to take now are integrating both sides (the
left-hand-side is instant, using the Fundamental Theorem of Calculus, but the right-hand-side might need a
little work), and then solving for y(t). Integrating both sides, we have
y(t)e p(t)dt
= e p(t)dt
g(t)dt.
Finally, solving for y(t),
y(t) = e− p(t)dt
e p(t)dt
g(t)dt. (4)
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2. And we are done! Outlined below is the algorithm for solving linear, first-order, ordinary differential equa-
tions with the integrating factor method.
To Solve an ODE with the Integrating Factor Method:
1. Put the equation in standard form such as in Eq. (1).
2. Calculate the integrating factor µ(t) using the definition in Eq. (2).
3. Multiply the differential equation by µ(t) and write it as written in Eq. (3).
4. Integrate both sides of the differential equation.
5. Solve for y(t) to obtain a general solution.
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