Welcome to the World of Differential Equations!

Hi there! Today we are going to explore a topic that sounds very fancy but is actually something you have already started doing: First Order Differential Equations. In your Pure Mathematics 1 (P1) course, this is the part where we use our integration skills to "reverse" a derivative and find the original equation of a curve.

Think of it like being a mathematical detective. If someone tells you how fast a car is moving (the derivative), can you figure out exactly where the car started and what path it took (the original equation)? That is exactly what we are doing here!

1. What is a First Order Differential Equation?

A differential equation is simply any equation that contains a derivative, like \( \frac{dy}{dx} \). The "First Order" part just means the highest derivative in the equation is the first derivative (no \( \frac{d^2y}{dx^2} \) allowed here!).

In your P1 syllabus (Section 5.2), you are required to solve equations that look like this:
\( \frac{dy}{dx} = f(x) \)

To solve this, we need to find \( y \). Since we know that integration is the reverse of differentiation, we integrate the function \( f(x) \) with respect to \( x \).

Analogy: Imagine differentiation is like "shredding" a document. Integration is like "taping" it back together. A differential equation is the shredded pile, and your job is to reconstruct the original page!

Quick Review: The Golden Rule of Integration
Don't forget the power rule! To integrate \( x^n \):
1. Add 1 to the power: \( n + 1 \)
2. Divide by the new power: \( \frac{1}{n+1} \)
3. Always add the Constant of Integration \( + C \)!

2. The "General Solution" and the Mystery of \( + C \)

When you integrate \( \frac{dy}{dx} \), you get what we call the General Solution. It always includes a \( + C \).

Why do we need \( + C \)? Because when we differentiate a constant (like 5, 10, or -100), it disappears and becomes zero. When we go backwards, we don't know what that original number was, so we use \( C \) as a placeholder.

Example:
If \( \frac{dy}{dx} = 3x^2 \), then:
\( y = \int 3x^2 dx \)
\( y = x^3 + C \)

This \( y = x^3 + C \) represents a whole "family" of curves that are all the same shape but shifted up or down on the graph. They all have the same gradient function, but different starting points.

Key Takeaway: The General Solution is an equation for \( y \) that still has the unknown constant \( C \) in it.

3. Finding the "Particular Solution"

Don't worry if the general solution feels a bit unfinished. Usually, an exam question will give you a specific point that the curve passes through, like \( (2, 10) \). These are called boundary conditions or initial conditions.

Using this point, we can solve for \( C \). Once we find the value of \( C \), we have the Particular Solution—the one, exact curve that fits the description.

Step-by-Step Guide to Finding the Equation of a Curve:

1. Integrate: Take the gradient function \( \frac{dy}{dx} \) and integrate it to find \( y = ... + C \).
2. Substitute: Plug in the \( x \) and \( y \) values from the given point into your new equation.
3. Solve: Find the value of \( C \).
4. Rewrite: Write down the final equation with the value of \( C \) included.

Example Problem:
Find the equation of the curve where \( \frac{dy}{dx} = 4x - 3 \) and the curve passes through the point \( (2, 5) \).

Step 1: Integrate
\( y = \int (4x - 3) dx \)
\( y = 2x^2 - 3x + C \)

Step 2: Substitute \( x=2 \) and \( y=5 \)
\( 5 = 2(2)^2 - 3(2) + C \)
\( 5 = 8 - 6 + C \)
\( 5 = 2 + C \)

Step 3: Solve for \( C \)
\( C = 3 \)

Step 4: Rewrite
The particular solution is \( y = 2x^2 - 3x + 3 \).

4. Common Pitfalls to Avoid

Even the best mathematicians make mistakes! Here are a few things to watch out for:

  • The Missing \( + C \): This is the most common mistake. If you forget \( + C \), you can't find the particular solution, and you'll lose easy marks!
  • Incorrect Integration: Remember that \( \frac{1}{x^2} \) should be written as \( x^{-2} \) before you try to integrate it.
  • Arithmetic Slips: When substituting negative numbers into your equation to find \( C \), use brackets on your calculator to avoid sign errors.

Did you know? Differential equations are used by scientists to predict how populations of animals grow, how diseases spread, and even how heat moves through a cup of coffee!

5. Summary Checklist

Before you tackle some practice questions, make sure you're comfortable with these points:

  • Can I integrate functions in the form \( x^n \)? (Remember: \( n \neq -1 \) for P1).
  • Do I understand that \( \frac{dy}{dx} \) represents the gradient of the curve?
  • Do I always include \( + C \) when finding a General Solution?
  • Can I substitute a point \( (x, y) \) into an equation to solve for \( C \)?

Encouraging Phrase: Don't worry if this seems tricky at first! Solving differential equations is just like solving a puzzle. Once you find that \( + C \), everything else falls into place. Keep practicing!

Quick Review Box:
Gradient Function: \( \frac{dy}{dx} \)
General Solution: \( y = f(x) + C \)
Particular Solution: The equation with a specific number for \( C \).
The goal: To find the original equation of the curve.