Integrating Factor
By Susam Pal on 10 Nov 2021
Introduction
One of the many techniques for solving ordinary differential
equations involves using an integrating factor. An
integrating factor is a function that a differential equation is
multiplied by to simplify it and make it integrable. It almost
appears to work like magic!
The Method
Let us first see how the integrating factor method works. In this
post, we will work with linear first-order ordinary differential
equations of type
to discuss, reason about, and illustrate this method. We will also
often use the Leibniz's notation and the Lagrange's
notation or simply interchangeably as is
typical in calculus. They all mean the same thing: the derivative
of the function with respect to Thus the above
differential equation may also be written as
Given a differential equation of this form, we first find an
integrating factor using the formula
Then we multiply both sides of the differential equation by this
integrating factor. Now remarkably, the left-hand side (LHS)
reduces to a single term consisting only of a derivative. As a
result, we can get rid of that derivative by integrating both sides
of the equation and we then proceed to obtain a solution.
An Example
Here is an example that demonstrates the method of using an
integrating factor. Let us say we want to solve the differential
equation
Indeed this is in the form with and We first obtain the integrating
factor
Now we multiply both sides of the differential equation by this
integrating factor and get
The LHS can now be simplified to This
can be verified using the product rule for derivatives. This
simplification of the LHS is the remarkable feature of this method.
Therefore the above equation can be written as
Note that the expression on the LHS is a product of the function and the integrating factor We will discuss this
observation in more detail a little later. Let us first complete
solving this differential equation. Since the LHS is now a single
term that consists of a derivative, obtaining a solution now simply
involves integrating both sides with respect to
Integrating both sides we get
where is the constant of integration. Finally, we divide
both sides by the integrating factor to get
We have now obtained a solution for the differential equation. If
we review the steps above, we will find that after multiplying both
sides of the given differential equation by the integrating factor,
the differential equation becomes significantly simpler and
integrable. In fact, after multiplying both sides of the given
differential equation by the integrating factor, the LHS always
becomes the derivative of the product of the function and
the integrating factor. We will now see why this is so.
An Interesting Relationship
Consider once again the linear first-order differential equation
We first find the integrating factor
The integrating factor obtained like this satisfies an interesting
relationship:
We can prove this relationship easily by differentiating both sides
of as follows:
Note that we use the chain rule to work out the derivative above.
This beautiful result is due to how the derivative of the
exponential function works. When we apply the chain rule to obtain
the derivative of we get
This nice property of the exponential function leads to the
interesting relationship in .
Simplification of LHS
Now let us multiply both sides of the differential equation
by the integrating factor By doing so,
we get
But from we know that so the above equation can be written as
Look what we have got on the LHS! We have the expansion of on the LHS. By product rule of
differentiation, we have Therefore the above equation can be written as
The "magic" has occurred here! Multiplying both sides of the
differential equation by the integrating factor has led us to an
equation that has got a single derivative only on the LHS. As a
result, finding the solution is now a simple matter of integrating
both sides, i.e.,
Thus
Note that the result of indefinite integral on the RHS will contain
the constant of integration, which we will denote as so the
final solution looks like
Illustration
Let us illustrate the method and its magic with a very simple
differential equation:
First we note that this equation is in the form with and We then find the
integrating factor
Then we multiply both sides of the differential equation by the
integrating factor to get
Now indeed the LHS can be written down as a single derivative as
shown below:
Note that the LHS is the derivative of the product of and
the integrating factor This is exactly what we discussed
in the previous section. We integrate both sides of the above
equation to get
Finally we divide both sides by the integrating factor to
get
We have arrived at the solution for the differential
equation.
Conclusion
In this post, we used very simple and convenient differential
equations that led to nice closed-form solutions. In practice,
differential equations can be quite complicated and may not always
lead to closed-form solutions. In such cases, we leave the result
in the form of an expression that contains an unsolved integral.
Such solutions may resemble the form shown in
.
The method of using integrating factors to solve differential
equations can also be extended to linear higher-order differential
equations. That is something we did not discuss in this post.
However, I hope that the intuition gained from understanding how and
why this method works for linear first-order differential equations
will be useful while studying such extensions of this method.