Introduction: Solving the Mystery of Change
Welcome to one of the most powerful tools in mathematics! So far, you have learned how to integrate functions to find areas. Now, we are going to use integration to solve differential equations.
Think of a standard equation like a photograph—it tells you exactly where something is at a specific moment. A differential equation is more like a rule for a movie—it describes how something changes over time. Whether it's a population of bees growing, a cup of coffee cooling down, or a car accelerating, differential equations are the language of the real world. In this chapter, we will learn how to "reverse" these rules of change to find the original formula.
Section 1: What is a Separable Differential Equation?
A first-order differential equation is simply an equation that involves a derivative, \( \frac{dy}{dx} \).
We say a differential equation is separable if we can "sort" the variables. This means we can move all the \( y \) terms to one side of the equals sign and all the \( x \) symbols to the other side.
Mathematically, it looks like this:
\( \frac{dy}{dx} = g(x)f(y) \)
The "Sorting Laundry" Analogy
Imagine you have a basket of mixed laundry (the equation). To wash them properly, you need to put the whites in one pile and the colors in another. Separating variables is just like that: we want all the \( y \)'s with the \( dy \) and all the \( x \)'s with the \( dx \).
Quick Review:
- If you can write the equation as \( \frac{1}{f(y)} dy = g(x) dx \), it is separable!
- Common Mistake: You cannot separate variables if they are added together in a way that can't be factored (e.g., \( \frac{dy}{dx} = x + y \) is NOT separable using this method).
Key Takeaway: Before you can integrate, you must ensure the \( dy \) is multiplied by terms containing only \( y \), and the \( dx \) is multiplied by terms containing only \( x \).
Section 2: The Step-by-Step Method
Don't worry if this seems tricky at first! Just follow these four steps every time:
Step 1: Separate the Variables
Move the \( dx \) to the right and any \( y \) terms to the left.
Example: If \( \frac{dy}{dx} = \frac{x}{y} \), then multiplying by \( y \) and \( dx \) gives: \( y \ dy = x \ dx \).
Step 2: Integrate Both Sides
Add the integration symbol to both sides:
\( \int y \ dy = \int x \ dx \)
Step 3: Add the Constant of Integration (\( + C \))
Crucial Point: You only need to add \( + C \) to one side of the equation (usually the \( x \) side). This represents the General Solution.
Step 4: Solve for \( y \) (if possible)
Rearrange the equation to get \( y = ... \). This is called an explicit solution.
Did you know?
The constant \( C \) is incredibly important. Without it, you are only finding one possible path. With \( C \), you are finding the "family" of all possible paths!
Section 3: General vs. Particular Solutions
When you solve a differential equation, you usually start with a General Solution (the one with the \( + C \)). However, if you are given a specific starting point, you can find the Particular Solution.
Finding the Particular Solution
If the question says the curve passes through a point (e.g., \( x = 0, y = 2 \)), these are your initial conditions.
- Find the General Solution first.
- Substitute the given values of \( x \) and \( y \) into your solution.
- Calculate the specific value of \( C \).
- Rewrite the equation using that value of \( C \).
Memory Aid:
- General = Generally speaking (contains \( C \)).
- Particular = This particular one (find the number for \( C \)).
Key Takeaway: Use initial conditions as soon as you have finished the integration to make the algebra easier!
Section 4: Factorisation - The Secret Skill
Sometimes, the variables don't look separable at first glance. The OCR syllabus (1.08k) mentions that you might need to factorise first.
Example: \( \frac{dy}{dx} = xy + 3x \)
This doesn't look separable yet because of the \( + \) sign. But if we factor out the \( x \):
\( \frac{dy}{dx} = x(y + 3) \)
Now it is separable! We can move \( (y+3) \) to the left and \( dx \) to the right:
\( \frac{1}{y+3} \ dy = x \ dx \)
Key Takeaway: If you see a mix of \( x \) and \( y \) added together, always look for a common factor to pull out.
Section 5: Modelling and Interpretation
OCR expects you to interpret these solutions in real-world contexts (1.08l), such as population growth or kinematics.
Common Models
- Exponential Growth: \( \frac{dP}{dt} = kP \). The rate of growth is proportional to the current population. This always leads to a solution involving \( e^{kt} \).
- Velocity/Kinematics: If you are given an equation for acceleration \( a = \frac{dv}{dt} \), you can separate variables to find the velocity \( v \).
Identifying Limitations
In the real world, math has limits! If a model predicts a population will reach 10 trillion in two days, the limitation might be a lack of food or space. Always ask: "Does this answer make sense for very large values of time?"
Example from Syllabus:
If a parachutist's velocity is \( v = 20 - 20e^{-t} \):
- As time \( t \) gets very large, \( e^{-t} \) goes to 0.
- Therefore, the velocity \( v \) approaches 20. This is the terminal velocity.
Quick Summary & Checklist
Before you finish, make sure you can:- Separate variables by multiplying or dividing (never adding or subtracting!).
- Factorise expressions to make them separable.
- Integrate correctly (remembering \( \ln|y| \) is common when integrating \( \frac{1}{y} \)).
- Always include \( + C \) immediately after integrating.
- Substitute values to find the Particular Solution.
- Explain what happens to the solution as time increases (limitations).
Top Tip: Keep your work neat! Differential equations often involve many steps of algebra, and most marks are lost through simple sign errors during rearranging.