Differential equations can look intimidating at first glance—symbols, derivatives, and variables everywhere. But in essence, they describe one of the most powerful ideas in mathematics: how quantities change over time. Understanding them doesn’t require complex theory—just clear reasoning and structured notes.
This guide explains differential equations in simple terms using RevisionDojo’s Notes feature to help you connect concepts, examples, and real-world meaning in a way that makes sense for IB Math.
Quick Start Checklist
Before diving into differential equations, make sure you:
- Understand basic differentiation and integration.
- Use RevisionDojo’s Notes to organize examples and step sequences.
- Identify real-life processes that involve change (growth, decay, motion).
- Learn to separate variables when possible.
- Reflect on the meaning of every equation’s result.
Differential equations tell the story behind the formulas.
Step 1: What Is a Differential Equation?
A differential equation is an equation that involves a function and its derivative.
Example:
dy/dx = 3x²
Here, dy/dx represents how y changes with x. Solving the equation means finding the function y that fits this relationship.
It’s essentially the reverse of differentiation—you’re finding the original function from its rate of change.
Step 2: Recognize the Real-World Meaning
Differential equations model change everywhere:
- Population growth → dy/dt = ky
- Cooling of an object → dT/dt = –k(T – A)
- Velocity and acceleration → dv/dt = a
These describe how something evolves over time or across another variable.
Step 3: Start With the Simplest Type — Separable Equations
A separable differential equation can be written so all y terms are on one side and all x terms on the other.
Example:
dy/dx = xy
Separate:
(1/y) dy = x dx
Integrate both sides:
∫(1/y) dy = ∫x dx → ln|y| = x²/2 + C
Finally, solve for y:
y = e^(x²/2 + C) = Ce^(x²/2)
Separation and integration are the backbone of most IB Math differential equation problems.
Step 4: Understand the Constant of Integration
Every differential equation includes an unknown constant (C). It represents a family of solutions.
When you’re given an initial condition like “y = 2 when x = 0,” you can solve for C specifically.
This turns the general solution into a particular solution that fits one scenario.
Step 5: Practice Common Equation Types
IB Math focuses on a few major forms:
- Exponential growth/decay: dy/dx = ky → y = Ae^(kx)
- Velocity and motion: dv/dt = a → integrate to find v(t).
- Cooling or heating: dT/dt = –k(T – A).
- Logistic growth (HL): dy/dt = ky(1 – y/L).
Add examples of each in your Notes so you can quickly recognize patterns.
Step 6: Interpret Graphs and Behavior
Once you solve, interpret what the solution tells you:
- Is the graph increasing or decreasing?
- Does it approach a limit?
- What does the constant control (rate, height, or shape)?
Connecting algebraic and visual meaning makes the concept stick.
Step 7: Check Solutions by Differentiating
You can always verify your solution by differentiating it and substituting back into the original equation.
If it works, you’ve solved correctly.
Example:
If y = e^(3x), dy/dx = 3e^(3x). Since dy/dx = 3y, the equation is satisfied.
Verification builds confidence and reinforces accuracy.
Step 8: Note Down Step Templates
In your Notes, write short “solution templates” for common types of differential equations. For instance:
Template for dy/dx = ky:
- Separate variables → (1/y) dy = k dx.
- Integrate both sides → ln|y| = kx + C.
- Simplify → y = Ae^(kx).
Having these outlines ready saves time in revision and exams.
Step 9: Relate to Real-Life Contexts
Many IB problems link differential equations to contexts like:
- Medicine (drug concentration in the body).
- Finance (compound interest).
- Physics (falling objects).
Write short examples in your notes to connect math to meaning—this helps both conceptual memory and IA inspiration.
Step 10: Reflect on Meaning, Not Just Method
After each problem, ask yourself:
- What does the equation represent physically?
- How does the constant affect the outcome?
- How would the model change if conditions changed?
Understanding why the equation behaves as it does is what IB examiners look for in full-mark responses.
Using Notes to Master Differential Equations
RevisionDojo’s Notes help you:
- Break complex calculus topics into simple explanations.
- Create step-by-step solution templates.
- Record solved examples and interpretations.
- Reflect on what each equation means conceptually.
- Review efficiently before exams.
It turns differential equations into approachable stories of change.
Common Mistakes to Avoid
Avoid these pitfalls when solving differential equations:
- Forgetting to separate variables properly. Always isolate y and x.
- Missing the constant of integration. Every integral includes +C.
- Not applying initial conditions. This gives the specific solution.
- Ignoring context. Real-world meaning adds clarity.
- Failing to check solutions. Verification ensures correctness.
Precision and reflection prevent confusion later.
Reflection: Change Is the Heart of Calculus
Differential equations show how small changes create large patterns. They connect theory to real life—growth, motion, heat, and more. Once you understand their structure, you’ll see calculus not as computation but as storytelling through change.
Frequently Asked Questions (FAQ)
1. What’s the difference between a differential and algebraic equation?
A differential equation involves derivatives—it describes how something changes, not just what it equals.
2. Why are differential equations important in IB Math?
They connect mathematics to real-world modeling, especially in HL topics and IAs.
3. How can I tell if an equation is separable?
If you can move all y terms to one side and all x terms to the other, it’s separable.
4. How do I know if my solution is correct?
Differentiate your answer and substitute it back into the original equation.
5. Should I memorize formulas or methods?
Understand patterns instead—most IB equations follow predictable structures.
Conclusion
Differential equations are simply stories of change written in symbols. When you break them down step by step, their meaning becomes clear and manageable.
Using RevisionDojo’s Notes, you can simplify, visualize, and internalize every differential concept—transforming a once-intimidating topic into one of your greatest IB Math strengths.
RevisionDojo Call to Action:
Turn complexity into clarity. Use RevisionDojo’s Notes to understand differential equations simply and build confidence in every calculus challenge.
