Integrating Factors: Definition and Purpose

The Problem: Why Standard Integration Fails

Let's start with our standard form equation:

You're familiar with solving separable equations, where we could move all the $y$ terms to one side and all the $x$ terms to the other, and then integrate both sides. Here, the term $p(x)y$ prevents us from doing that. We can't simply integrate the equation term by term, because we don't know how to compute $\int (y^{\prime }+p(x)y)dx$ directly.

The goal is to manipulate the left-hand side, $y^{\prime }+p(x)y$, into a form that we can easily integrate. The trick lies in recognizing its similarity to the result of the product rule from calculus, $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$.

The Solution: The Integrating Factor

The strategy is to multiply our entire differential equation by a special, currently unknown function, which we'll call $\mu (x)$. This function, the integrating factor, is specifically chosen to force the left-hand side to become the derivative of a product.

Let's see how this works. Multiplying the standard form by $\mu (x)$ gives:

Our goal is to make the left-hand side the result of the product rule for ${\left(\mu (x)y\right)}^{\prime }$. Let's expand that derivative:

Now, compare what we have with what we want:

• Have: $\mu (x)y^{\prime }+\mu (x)p(x)y$
• Want: $\mu (x)y^{\prime }+{\mu }^{\prime }(x)y$

The $\mu (x)y^{\prime }$ terms match perfectly. For both expressions to be equal, the other terms must also be equal:

Dividing by $y$, we find the condition that our integrating factor $\mu (x)$ must satisfy:

This is the fundamental property that defines the integrating factor.

The following video provides an excellent and clear derivation of this concept.

Formal Definition and Purpose

Let's formally summarize what we've learned from the derivation.

• Purpose: The purpose of an integrating factor $\mu (x)$ is to transform the left-hand side of a first-order linear ODE from the form $y^{\prime }+p(x)y$ into the derivative of a product, $\frac{d}{dx}[\mu (x)y]$. This makes the differential equation directly integrable.

• Definition: An integrating factor is a function $\mu (x)$ that satisfies the condition ${\mu }^{\prime }(x)=\mu (x)p(x)$. By solving this simple separable differential equation for $\mu (x)$, we arrive at its general formula.

As shown in the video, solving $\frac{d\mu }{dx}=\mu \cdot p(x)$ gives:

Note that we can ignore the constant of integration here. Since our goal is to find any function that satisfies the condition, we can choose the simplest one where the constant is zero (or, more formally, the multiplicative constant is 1).

The following reading from Paul's Online Math Notes provides a textual walkthrough of the same logic, which can help solidify your understanding.

Conclusion

Today, we've explored the theory behind one of the most elegant techniques for solving differential equations. We've moved from simply identifying the parts of an equation to understanding the "why" behind the solution method.

Key Takeaways:

• A first-order linear equation $y^{\prime }+p(x)y=q(x)$ cannot be integrated directly.
• An integrating factor, $\mu (x)$, is a function we multiply the equation by to make the left-hand side the derivative of a product, ${\left(\mu (x)y\right)}^{\prime }$.
• This integrating factor is derived from $p(x)$ and is given by the formula $\mu (x)=e^{\int p(x)dx}$.

Preview of the next lesson:
We have now established what the integrating factor is and why it works. In the next lesson, we will put this theory into action. We will use the integrating factor to derive the full general solution for a first-order linear equation and begin solving concrete problems step-by-step.