First order differential equations

Welcome to First Order Differential Equations!

In your previous math studies, you’ve spent a lot of time solving equations to find a specific number (like \(x = 5\)). In this chapter, we are leveling up! We are going to solve Differential Equations (DEs), where the "answer" isn't a single number, but an entire function.

Differential equations are the language of the universe. They describe how things change—from how a cup of tea cools down to how a population of rabbits grows. Don't worry if this seems tricky at first; we will break it down into three clear methods used in the FP2 syllabus.

1. Separation of Variables

This is likely a technique you’ve seen before in Pure Mathematics, but in FP2, we take it a step further. The goal is simple: get all the \(y\) terms on one side with \(dy\), and all the \(x\) terms on the other side with \(dx\).

The Process:

1. Rearrange the equation into the form \(f(y) dy = g(x) dx\).
2. Integrate both sides: \(\int f(y) dy = \int g(x) dx\).
3. Add the Constant (\(+C\)): This is the most common mistake! Always add the constant of integration as soon as you integrate.
4. Solve for \(y\) if possible to get an explicit solution.

Example:

Solve \(\frac{dy}{dx} = \frac{x}{y}\).
Separate: \(y dy = x dx\)
Integrate: \(\int y dy = \int x dx\)
Result: \(\frac{1}{2}y^2 = \frac{1}{2}x^2 + C\)

Quick Review: To find a Particular Solution, you will be given "boundary conditions" (like \(y=2\) when \(x=0\)). Plug these in at the very end to find the specific value of \(C\).

Key Takeaway: If you can move the \(x\)'s and \(y\)'s to opposite sides of the equals sign, use Separation of Variables!

2. First Order Linear Equations (The Integrating Factor)

Sometimes, you cannot separate the variables. If your equation looks like this:
\(\frac{dy}{dx} + Py = Q\)
(where \(P\) and \(Q\) are functions of \(x\)), you need a special tool called an Integrating Factor.

The "Secret Sauce": The Integrating Factor (IF)

The Integrating Factor is defined as: \(IF = e^{\int P dx}\)

Step-by-Step Method:

1. Standard Form: Make sure the coefficient of \(\frac{dy}{dx}\) is exactly 1. If it isn't, divide the whole equation by whatever is in front of it.
2. Find \(P\): Identify what is multiplying the \(y\).
3. Calculate the IF: Find \(e^{\int P dx}\). (Note: Usually, the \(\ln\) and \(e\) will cancel out, leaving you with a simple expression).
4. Multiply: Multiply every single term in the equation by your IF.
5. The Reverse Product Rule: The left side of your equation now magically becomes \(\frac{d}{dx}(y \cdot IF)\).
6. Integrate: Integrate both sides with respect to \(x\): \(y \cdot IF = \int (Q \cdot IF) dx\).
7. Solve: Divide by the IF to find \(y\).

Common Mistake: Forgetting to divide by the coefficient of \(\frac{dy}{dx}\) at the very start. If you don't start in Standard Form, the whole method fails!

Did you know? This method is like finding the missing piece of a puzzle that completes the Product Rule on the left-hand side of the equation.

Key Takeaway: For linear equations, find the Integrating Factor \(e^{\int P dx}\), multiply everything by it, and then integrate.

3. Reducible Differential Equations (Substitution)

Some equations look terrifying and don't fit either of the patterns above. However, we can use a Substitution to transform them into a form we know how to solve.

The Good News:

In your FP2 exam, the question will almost always tell you which substitution to use! For example: "Use the substitution \(z = y^{-2}\) to show that..."

How to handle substitutions:

1. Differentiate the substitution: If you are given \(z = f(y)\), differentiate it with respect to \(x\) to find \(\frac{dz}{dx}\). You will likely need to use the Chain Rule (e.g., \(\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}\)).
2. Substitute: Replace all instances of \(y\) and \(\frac{dy}{dx}\) in the original equation with expressions involving \(z\) and \(\frac{dz}{dx}\).
3. Solve: The new equation will usually be a simple linear DE (use Method 2) or separable (use Method 1).
4. Substitute back: Once you have a solution for \(z\), replace \(z\) with the original \(y\) expression to finish.

Encouragement: Substitution is just "mathematical dress-up." We change the clothes of the equation to make it look familiar, solve it, and then change the clothes back at the end!

Key Takeaway: Use the Chain Rule to change the variables, solve the simpler equation, and don't forget to switch back to the original variables at the end.

4. Families of Curves

When we solve a DE and end up with a constant \(+C\), we call this the General Solution. Because \(C\) can be any number, the general solution represents an infinite family of curves.

Sketching the Curves:

You may be asked to sketch a few members of this family. Each value of \(C\) gives a different curve. Usually, these curves follow a similar shape but are shifted or scaled.
Analogy: Think of a general solution as a "template." If the template is a circle, the family of curves is a set of concentric circles (like ripples in a pond) representing different values of \(C\).

Quick Tip: Particular solutions (where you've found a specific \(C\)) are just one single line/curve within that entire family.

Key Takeaway: The General Solution is the "whole family," while the Particular Solution is just "one person" in that family.

Summary Checklist for Success

• Separable? Get \(x\) on one side, \(y\) on the other.
• Linear? Use \(IF = e^{\int P dx}\). Remember Standard Form first!
• Substitutions? Use the Chain Rule carefully to swap variables.
• Constant \(+C\)? Always add it immediately after integrating.
• Boundary Conditions? Use them to find a Particular Solution.