Differential equations

Welcome to Differential Equations!

In your mathematical journey so far, you have spent a lot of time solving equations to find a specific number, like x = 5. In this chapter, we are stepping up a level. We are going to learn how to solve differential equations. Instead of finding a single number, we are looking for an entire function (like \( y = x^2 + 3 \)) that describes how something changes over time or space.

Differential equations are the language of the universe. They describe how populations grow, how diseases spread, and even how a cup of tea cools down. Don't worry if this seems tricky at first—once you learn the pattern of "separating" the variables, it becomes much more manageable!

1. Formulating Differential Equations

The first step is often translating a real-world situation into a mathematical equation. This is called formulating a differential equation. The key is to look for the phrase "rate of change". This always signals a derivative, usually with respect to time, written as \( \frac{dy}{dt} \).

Key Concepts in Formulation

• Direct Proportionality: If the rate of change of \( y \) is proportional to \( y \), we write: \( \frac{dy}{dt} = ky \).
• Inverse Proportionality: If the rate of change of \( y \) is inversely proportional to \( x \), we write: \( \frac{dy}{dx} = \frac{k}{x} \).
• Growth and Decay: If something is increasing, \( k \) is positive. If it is decreasing (like a radioactive substance decaying), \( k \) is negative.

Example: Population Growth

A population of rabbits, \( P \), grows at a rate proportional to the current population.
This translates to: \( \frac{dP}{dt} = kP \)

Quick Review Box: The Constant of Proportionality

Always remember to include the constant \( k \) when you see the word "proportional." It represents the specific rate for that particular situation.

Section Takeaway: When formulating, identify the rate of change (\( \frac{dy}{dt} \)), identify what it depends on, and connect them with a constant \( k \).

2. Solving by Separating Variables

The syllabus for MEI H640 focuses on first-order differential equations that can be solved by separating variables. Think of this like sorting your laundry—you want all the \( y \) terms on one side and all the \( x \) (or \( t \)) terms on the other.

Step-by-Step: The Separation Method

If you have an equation like \( \frac{dy}{dx} = g(x)f(y) \), follow these steps:

Step 1: Separate. Move all \( y \) terms to the side with \( dy \) and all \( x \) terms to the side with \( dx \).
\( \frac{1}{f(y)} dy = g(x) dx \)

Step 2: Integrate. Add integral signs to both sides.
\( \int \frac{1}{f(y)} dy = \int g(x) dx \)

Step 3: Solve. Perform the integration. Crucial: Add a constant of integration \( + C \) to one side (usually the \( x \) side).

Step 4: Rearrange. If possible, rearrange the equation to make \( y \) the subject.

Did you know?

The constant \( + C \) is the most forgotten part of A Level Maths! Without it, you only find one possible answer instead of the whole family of curves that could solve the equation.

General vs. Particular Solutions

• General Solution: The answer that still contains the \( + C \). It represents all possible solutions.
• Particular Solution: If you are given initial conditions (e.g., "when \( x = 0, y = 2 \)"), you can plug these in to find the exact value of \( C \).

Common Mistake to Avoid:

When you have \( \int \frac{1}{y} dy = \ln|y| \), and you eventually exponentiate both sides to get rid of the log, remember that \( e^{\dots + C} \) becomes \( Ae^{\dots} \) where \( A = e^C \). This is a much cleaner way to write your final answer!

Section Takeaway: Separating variables is just algebra followed by integration. Always include \( + C \) immediately after integrating.

3. Interpreting and Limitations

Once you have solved the equation, you need to understand what it tells you about the real world. This is especially important in kinematics (motion) and modelling.

Contextual Meaning

If your solution describes the temperature of a cake cooling down, and your model predicts the temperature will eventually reach \( -100^{\circ}C \) in a warm kitchen, you know something is wrong! Mathematics is powerful, but it must make sense.

Identifying Limitations

Models are simplifications. When asked to "identify limitations," consider:
• The Domain: Does the model work for all time (\( t \ge 0 \)) or only for a short period?
• Unrealistic Growth: Models like \( \frac{dP}{dt} = kP \) suggest a population will grow to infinity, which isn't possible due to food and space limits.
• Assumptions: Did we assume air resistance is zero? Did we assume the rate of growth stays constant?

Memory Aid: The "Reality Check"

When interpreting, ask yourself: "What happens as \( t \to \infty \)? Does this value stay sensible?"

Section Takeaway: A solution is only as good as its model. Always check if the math matches the physical reality of the problem.

Final Summary of Differential Equations

• Formulate: Turn words like "rate of change" into derivatives and proportionality statements.
• Separate: Get all \( y \)s with \( dy \) and all \( x \)s with \( dx \).
• Integrate: Don't forget the \( + C \)!
• Particularize: Use given coordinates to find the specific value of \( C \).
• Evaluate: Check if your solution makes sense for the real-world scenario provided.

Differential equations can feel like a lot of steps, but it's just combining your skills in Algebra, Differentiation, and Integration into one "super-topic." Keep practicing the separation step, and the rest will fall into place!