Integrating Factors for First-Order Linear Equations

The Full Derivation

Let's begin where we left off, with the two key pieces of information we have:

1. The standard form of a first-order linear ODE: $\frac{dy}{dx}+p(x)y=q(x)$
2. The integrating factor: $\mu (x)=e^{\int p(x)dx}$

Our objective is to find a general formula for $y(x)$. The derivation follows a logical, step-by-step process.

To see this derivation in detail, let's turn to a clear textual explanation.

Let's break down the key steps from that reading.

Step 1: Multiply by the Integrating Factor

We start with the standard form and multiply every term by $\mu (x)$:

Step 2: The "Magic" of the Product Rule

This is the crucial step. As we established in the last lesson, the integrating factor was specifically constructed so that the left-hand side of this new equation is exactly the derivative of the product $\mu (x)y$. Let's quickly verify this.

The product rule for $\frac{d}{dx}[\mu (x)y]$ is:

And from the previous lesson, we know that the defining property of $\mu (x)$ is $\frac{d\mu }{dx}=\mu (x)p(x)$. Substituting this in gives:

This is precisely the left side of our equation from Step 1. Therefore, we can substitute it, simplifying the entire differential equation to:

Step 3: Integrate Both Sides

Our equation is now in a very simple form. The left side is a derivative with respect to $x$, and the right side is a function of $x$. We can solve it by integrating both sides with respect to $x$:

By the Fundamental Theorem of Calculus, integrating the derivative on the left side simply gives us the original function back. This is where we must also introduce the constant of integration, $C$, as we are performing an indefinite integral:

A crucial point: In the previous lesson, we could ignore the constant of integration when finding $\mu (x)$ because we only needed one suitable function. Here, the constant $C$ is essential as it represents the family of all possible solutions to the ODE.

Step 4: Isolate y(x)

Finally, to get the general solution for $y(x)$, we just need to divide by $\mu (x)$:

And there we have it. This is the general solution to any first-order linear ordinary differential equation.

The following video provides a dynamic summary of this final part of the derivation.

The Solution Process Summarized

While memorizing the final formula is an option, it's often more reliable to remember the process, which is a core part of your request for a general mathematical understanding. Let's recap the entire procedure from start to finish.

To solve a first-order linear ODE:

1. Standard Form: Ensure the equation is in the form $\frac{dy}{dx}+p(x)y=q(x)$.
2. Find the Integrating Factor:
   • Identify $p(x)$.
   • Calculate $\mu (x)=e^{\int p(x)dx}$. (You can omit the constant of integration here).
3. Construct the Solution: The general solution is given by: $$y(x)=\frac{1}{\mu (x)}\left(\int \mu (x)q(x)dx+C\right)$$ (You must include the constant of integration $C$ here).

This three-step process is the "recipe" for solving any first-order linear equation. We have now fully derived and justified it.

Conclusion

In this lesson, we completed the theoretical foundation of the integrating factor method. We built upon the definition from the previous lesson to derive the full general solution for a first-order linear differential equation.

Key Takeaways:

• Multiplying the standard form equation by the integrating factor $\mu (x)$ transforms the left side into $\frac{d}{dx}[\mu (x)y]$.
• Integrating both sides of the transformed equation yields $\mu (x)y=\int \mu (x)q(x)dx+C$.
• The general solution is obtained by solving for $y(x)$: $$y(x)=\frac{1}{\mu (x)}\left(\int \mu (x)q(x)dx+C\right)$$

Preview of the next lesson:
We have now concluded the "theory" part of this module. In the next lesson, we will move to "practice." We will apply the solution process we've derived to solve several different first-order linear differential equations from start to finish.