Differential equations

Introduction to Differential Equations

Welcome to the world of Differential Equations! While the name sounds intimidating, a differential equation is simply an equation that involves a derivative (like \( \frac{dy}{dx} \)).

Think of a regular equation as a "snapshot" of a situation (e.g., "The car is at mile marker 50"). A differential equation is more like a "rule of motion" (e.g., "The car’s speed is doubling every hour"). In this chapter, you will learn how to take these rules of change and turn them back into normal equations. This is a vital skill used by scientists to predict population growth, by engineers to build bridges, and by economists to model stock markets!

1. What is a Differential Equation?

A differential equation (DE) is any equation that contains a derivative. For your Oxford AQA International AS Level, we focus on first-order differential equations. This just means the highest derivative in the equation is \( \frac{dy}{dx} \).

The Goal: When we "solve" a DE, we are looking for the original relationship between \( x \) and \( y \). We want to get rid of the \( \frac{dy}{dx} \) and find an equation that looks like \( y = f(x) \).

Analogy: Imagine you have a map that tells you the slope of the ground at every point. Solving the DE is like using that map to draw the actual path of the mountain.

Key Takeaway: Solving a differential equation is essentially integrating to find the original function.

2. Solving by Separating Variables

The main method you need to master is called Separation of Variables. This is used when you can pull all the \( y \)'s to one side of the equals sign and all the \( x \)'s to the other.

Step-by-Step Process:

1. Rearrange: Move all terms with \( y \) to the left side (with the \( dy \)) and all terms with \( x \) to the right side (with the \( dx \)). It should look like this:
\( g(y) dy = f(x) dx \)

2. Integrate: Put an integral sign in front of both sides:
\( \int g(y) dy = \int f(x) dx \)

3. Add the Constant: This is the most important step! Add a \( + C \) to the side with the \( x \).
Don't worry if this seems tricky at first; it becomes a habit with practice!

4. Solve for y: If possible, rearrange the final answer to get \( y = ... \)

Common Mistake to Avoid: Never leave \( dx \) or \( dy \) in the denominator. They must always be on the "top" of the fraction before you integrate.

Example: Solve \( \frac{dy}{dx} = 3x^2y \)
Step 1: Separate. Divide by \( y \) and multiply by \( dx \):
\( \frac{1}{y} dy = 3x^2 dx \)
Step 2: Integrate.
\( \int \frac{1}{y} dy = \int 3x^2 dx \)
Step 3: Result.
\( \ln|y| = x^3 + C \)

Key Takeaway: If you can't separate the \( x \)'s and \( y \)'s by multiplying or dividing, this method won't work!

3. General vs. Particular Solutions

When you solve a DE, you will encounter two types of answers:

General Solution: This is the answer that still contains the \( + C \). It represents a whole "family" of curves that all follow the same rule of change.

Particular Solution: This is a specific answer where we find the exact value of \( C \). To do this, we need Initial Conditions (also called boundary conditions) — basically, a specific point \( (x, y) \) that the curve passes through.

Memory Aid:
General = Generous (It leaves the \( C \) in!)
Particular = Precise (We find the exact number for \( C \)).

Quick Review Box: Finding C
1. Solve the DE to get the General Solution.
2. Plug in the given values for \( x \) and \( y \).
3. Solve for \( C \).
4. Rewrite the equation with the new number for \( C \).

Key Takeaway: You can only find the Particular Solution if the question gives you a specific point or set of values.

4. Modeling Growth and Decay

The syllabus often requires you to create a differential equation from a word problem. This usually involves Rates of Change.

Common Phrases:
• "The rate of increase of population \( P \) is proportional to the population": \( \frac{dP}{dt} = kP \)
• "The rate of decrease of a substance \( M \) is proportional to \( M \)": \( \frac{dM}{dt} = -kM \)

Did you know? The constant \( k \) is called the constant of proportionality. If something is growing, \( k \) is positive. If it is shrinking (like radioactive decay), \( k \) is negative.

Real-World Analogy: Think of your bank account. If the interest rate is fixed, the "rate of increase" of your money is proportional to how much money you already have. The more you have, the faster it grows! This is modeled by the DE \( \frac{dA}{dt} = rA \).

Key Takeaway: Always look for the word "proportional to" — it tells you exactly how to set up your equation!

Summary of Common Integrals for DEs

Since solving DEs requires integration, make sure you are comfortable with these basics from earlier chapters:

• \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)
• \( \int \frac{1}{x} dx = \ln|x| + C \)
• \( \int e^{ax} dx = \frac{1}{a} e^{ax} + C \)

Final Tip: When your equation involves \( \ln|y| \), you will often need to use exponential rules to clean it up. Remember that \( e^{\ln|y|} = y \). If you have \( \ln|y| = x + C \), you can write it as \( y = e^{x+C} \), which is the same as \( y = Ae^x \) (where \( A = e^C \)). This is a very common trick in exams!