Further Mathematics (9665) | Differential equations – first order

Ready to Master First-Order Differential Equations?

Welcome to one of the most powerful topics in Further Mathematics! Differential equations might seem intimidating, but they are simply the mathematical language used to describe change in the world—like how temperature cools, how current flows in a circuit, or how quickly medicine is absorbed by the body.

In this chapter, we focus on solving a specific type of equation: First-Order Linear Differential Equations. Don't worry if this seems tricky at first; we have a brilliant, step-by-step method (using the Integrating Factor) that always works!

1. Recognizing the Standard Form

A first-order linear differential equation is an equation involving the first derivative (\(\frac{dy}{dx}\)) and the function \(y\) itself, which can be rearranged into a special structure called the Standard Form.

The Key Standard Form

You must be able to recognize and rearrange any first-order linear DE into this exact structure:

\[ \frac{dy}{dx} + P(x)y = Q(x) \]

Here’s what you need to know about the parts:

• \(\frac{dy}{dx}\): This term must have a coefficient of 1. If it doesn't (e.g., if you have \(x\frac{dy}{dx}\)), you must divide the entire equation by the coefficient (\(x\)) first.
• \(P(x)\): This is the function of \(x\) (or a constant) that is multiplying \(y\).
• \(Q(x)\): This is the function of \(x\) (or a constant) on the right-hand side.

Quick Example: If you have \(x\frac{dy}{dx} - y = x^3\), you must divide by \(x\) to get the standard form:

\[ \frac{dy}{dx} + \left(-\frac{1}{x}\right)y = x^2 \]

In this case, \(P(x) = -\frac{1}{x}\) and \(Q(x) = x^2\).

2. The Integrating Factor (IF) – Your Magic Key

The core difficulty in solving \(\frac{dy}{dx} + Py = Q\) is that the left-hand side is generally not the derivative of a simple function. The Integrating Factor is a special function that, when multiplied throughout the equation, magically turns the left-hand side into the derivative of a product.

How to Find the Integrating Factor (IF)

The formula for the Integrating Factor is given by:

\[ \text{IF} = e^{\int P \, dx} \]

Important Notes on Calculating IF:

1. You only integrate \(P(x)\). You do not need to include the constant of integration (\(C\)) when calculating \(\int P \, dx\). We will handle the general constant later.
2. Remember that the integral of \(\frac{1}{x}\) is \(\ln|x|\). When using IF, the absolute value is often dropped, and we use the property \(e^{\ln f(x)} = f(x)\).

Did you know? This method works because multiplying by the IF reverses the Product Rule! When you multiply the DE by \(e^{\int P \, dx}\), the left side becomes exactly the derivative of \(y \times e^{\int P \, dx}\).

Step-by-Step Process for the Integrating Factor Method

Once you have the Standard Form, follow these six essential steps:

Step 1: Identify P(x)

Make sure your equation is \(\frac{dy}{dx} + P(x)y = Q(x)\).

Step 2: Calculate the Integrating Factor (IF)

Find \(\text{IF} = e^{\int P \, dx}\). Simplify this as much as possible.

Step 3: Multiply the DE by the IF

Multiply every single term in the standard form DE by the IF.

Step 4: Rewrite the Left-Hand Side (LHS)

The magic happens here! The entire LHS will always simplify to the derivative of the product of \(y\) and the IF:

\[ \frac{d}{dx} (y \times \text{IF}) = Q \times \text{IF} \]

Memory Aid: If you've calculated the IF correctly, the LHS must always fit this form. If it doesn't, check your algebra or integration of \(P(x)\).

Step 5: Integrate Both Sides

Integrate both sides of the equation with respect to \(x\).

\[ y \times \text{IF} = \int (Q \times \text{IF}) \, dx + C \]

CRITICAL POINT: You must include the constant of integration, \(C\), on the right-hand side. This constant gives you the General Solution.

Step 6: Solve for y

Divide by the IF to get the final solution for \(y\). This is the General Solution to the differential equation.

\[ y = \frac{1}{\text{IF}} \left[ \int (Q \times \text{IF}) \, dx + C \right] \]

3. Understanding the General Solution: CF and PI

The structure of your General Solution reveals two important components, as required by the syllabus: the Complementary Function (CF) and the Particular Integral (PI).

The General Solution is written as:

\[ y = y_{\text{CF}} + y_{\text{PI}} \]

The Complementary Function (\(y_{\text{CF}}\))

The CF is the part of the solution that contains the arbitrary constant \(C\).

• It represents the transient or "natural" behaviour of the system being modelled.
• It is found by solving the associated homogeneous equation (i.e., setting \(Q(x)=0\)).
• In the final General Solution (Step 6), the CF is the term involving \(C\).

The Particular Integral (\(y_{\text{PI}}\))

The PI is the part of the solution that contains no arbitrary constants.

• It represents a specific response to the input function \(Q(x)\).
• In the final General Solution (Step 6), the PI is the term that does not contain \(C\).

Analogy: Imagine throwing a ball. The CF describes how the ball naturally moves due to gravity (the system itself), and the PI describes the specific influence of any external force (like an air jet) being applied. The total motion is the sum of these two effects.

4. Finding Particular Solutions

The General Solution gives an infinite family of possible solutions (one for every possible value of \(C\)). In real-world problems, we usually need a Particular Solution that fits specific circumstances.

Using Boundary Values and Initial Conditions

To find the unique value of \(C\), you need an initial condition or boundary value. This is usually a specific point \((x_0, y_0)\) that the solution must pass through.

The Process:

1. Find the General Solution \(y(x)\) which includes the constant \(C\).
2. Substitute the given initial condition \((x_0, y_0)\) into the General Solution.
3. Solve the resulting algebraic equation to find the numerical value of \(C\).
4. Substitute this numerical value of \(C\) back into the General Solution to obtain the Particular Solution.

Common Mistake to Avoid: Do not wait until the end to integrate and find \(C\). The integration constant \(C\) must be included immediately in Step 5 (when you integrate the term \(Q \times \text{IF}\)) to ensure it is multiplied/divided by the correct functions of \(x\).

5. Final Checklist for Solving \(\frac{dy}{dx} + Py = Q\)

Use this checklist in your exams to ensure you haven't missed any vital steps. Keep practicing these steps, and you'll find these problems become routine!

1. Standard Form: Is the equation in the form \(\frac{dy}{dx} + P(x)y = Q(x)\)? (Check coefficient of \(\frac{dy}{dx}\) is 1).
2. P(x) and Q(x): Are P and Q correctly identified?
3. Integrating Factor: Is the IF calculated correctly as \(\text{IF} = e^{\int P \, dx}\)? (No \(C\)).
4. Product Rule Form: Is the equation rewritten as \(\frac{d}{dx} (y \times \text{IF}) = Q \times \text{IF}\)?
5. Integration: Is the right-hand side integrated correctly, and has \(+C\) been included?
6. General Solution: Is \(y\) isolated to give the General Solution? (This contains the CF and PI parts).
7. Particular Solution (if needed): Are the boundary conditions used to find the specific numerical value of \(C\)?