Differential equations C
📐 Differential Equations
Complete Theory & Formulas Guide
Cambridge AS & A Level Mathematics (Pure Mathematics 3)
⏱️ Reading Time: 30-40 minutes
What You'll Learn:
- What differential equations are
- How to solve them using separation of variables
- How to form differential equations from real-world problems
- Practical applications in physics, biology, and economics
1️⃣ What is a Differential Equation?
A differential equation is an equation that contains derivatives such as dy/dx or d²y/dx².
dy/dx = 3x²
🔍 Understanding the Difference
| Type | Example | Solution Type |
|---|---|---|
| Algebraic Equation | y = x³ + 5 | A function |
| Differential Equation | dy/dx = 3x² | A function (y = x³ + C) |
📊 Types of Solutions
1. General Solution
Contains an arbitrary constant C
General Solution: y = x³ + C
2. Particular Solution
Uses initial conditions to find specific value of C
5 = 0³ + C → C = 5
Particular Solution: y = x³ + 5
🌍 Real-World Applications
- Biology: Rate of bacteria growth
- Physics: Motion of particles, Newton's law of cooling
- Economics: Depreciation of car value
- Medicine: Drug concentration in bloodstream
- Engineering: Heat transfer, electrical circuits
2️⃣ The Separation of Variables Method
This is our main technique for solving first-order differential equations!
🎯 The Basic Idea
• All y terms are on one side
• All x terms are on the other side
Then integrate both sides separately
⭐ Key Principle
Meaning: Integrating dy/dx with respect to x gives you y back!
📝 General Method
Then: ∫f(y) dy = ∫g(x) dx
🔢 Step-by-Step Process
Get form: f(y) dy = g(x) dx
∫f(y) dy = ∫g(x) dx
Don't forget + C on one side!
Rearrange to get y = ...
Find value of C
3️⃣ Worked Examples
📌 Example 1: Simple Case
dy = 6(x + 1)² dx
∫dy = ∫6(x + 1)² dx
y = 6 × (x + 1)³/3 + C
y = 2(x + 1)³ + C
📌 Example 2: Variables Need Separating
dy/dx = xy
(1/y) dy/dx = x
(1/y) dy = x dx
∫(1/y) dy = ∫x dx
ln|y| = x²/2 + C
|y| = e^(x²/2 + C)
|y| = e^C × e^(x²/2)
Let D = e^C:
y = D × e^(x²/2)
📌 Example 3: With Initial Condition
2y dy/dx = x + 1
2y dy = (x + 1) dx
∫2y dy = ∫(x + 1) dx
y² = x²/2 + x + C
3² = 0²/2 + 0 + C
9 = C
4️⃣ Key Formulas & Patterns
🔑 Essential Integration Results
| Function | Integral |
|---|---|
| ∫(1/y) dy | ln|y| + C |
| ∫y dy | y²/2 + C |
| ∫y² dy | y³/3 + C |
| ∫y^n dy | y^(n+1)/(n+1) + C |
| ∫e^y dy | e^y + C |
| ∫x dx | x²/2 + C |
| ∫x^n dx | x^(n+1)/(n+1) + C |
| ∫e^x dx | e^x + C |
🎯 Common Differential Equation Types
dy/dx = f(x)
Solution: y = ∫f(x) dx + C
dy/dx = f(x)g(y)
Solution: ∫(1/g(y)) dy = ∫f(x) dx
dP/dt = kP (k > 0)
Solution: P = P₀e^(kt)
dP/dt = -kP (k > 0)
Solution: P = P₀e^(-kt)
5️⃣ Forming Differential Equations from Problems
📖 Translation Dictionary
| Words in Problem | Mathematical Symbol |
|---|---|
| Rate of increase | +dy/dt or +dx/dt |
| Rate of decrease | -dy/dt or -dx/dt |
| Is proportional to | = k × (something) |
| Inversely proportional to | = k / (something) |
| Rate of change | dy/dt (can be + or -) |
| Directly proportional | = k × (something) |
🎓 Translation Examples
"The rate of increase of bacteria is proportional to the number present"
"rate of increase" → dx/dt
"proportional to number present" → = kx
Result: dx/dt = kx
"Velocity is inversely proportional to displacement"
v = ds/dt
"inversely proportional to displacement" → = k/s
Result: ds/dt = k/s
"The rate of decrease of temperature is proportional to the difference between object and room temperature"
"rate of decrease" → -dT/dt
"proportional to difference" → = k(T - T₀)
Result: dT/dt = -k(T - T₀)
6️⃣ Important Models
📈 Model 1: Exponential Growth
General Solution: P = P₀e^(kt)
Where:
• P₀ = initial value at t = 0
• k = growth rate constant
• t = time
P increases rapidly as t increases
Used for:
- Bacterial population growth
- Compound interest
- Radioactive production
📉 Model 2: Exponential Decay
General Solution: P = P₀e^(-kt)
Where:
• P₀ = initial value at t = 0
• k = decay rate constant
• t = time
P decreases and approaches zero
Used for:
- Radioactive decay
- Drug concentration in blood
- Depreciation of car value
- Cooling of hot objects
🌡️ Model 3: Newton's Law of Cooling
"Rate of change of temperature is proportional to the difference between object temperature and surroundings"
Differential Equation: dT/dt = -k(T - Tₛ)
General Solution: T = Tₛ + (T₀ - Tₛ)e^(-kt)
Where:
• T = temperature of object
• Tₛ = temperature of surroundings (constant)
• T₀ = initial temperature at t = 0
• k = cooling constant (k > 0)
• t = time
Initial: T₀ = 90°C, Tₛ = 20°C
Eventually: T → 20°C (room temperature)
7️⃣ Problem-Solving Strategy
What is changing? With respect to what?
Use the translation dictionary
Write dy/dx = ... or dP/dt = ...
Get f(y) dy = g(x) dx
∫f(y) dy = ∫g(x) dx + C
Find C (and k if needed)
Rearrange to y = ... or x = ...
Calculate specific values, check units
8️⃣ Complete Worked Example
🎯 Full Problem
A motorbike's value decreases at a rate proportional to its current value. Initially worth $10,000. After 3 years, worth $5,000. Find value after 2 years.
V = value of motorbike ($)
t = time (years)
"decreases at rate proportional to value"
dV/dt = -kV (negative because decreasing)
(1/V) dV = -k dt
∫(1/V) dV = ∫-k dt
ln V = -kt + C
ln(10,000) = 0 + C
C = ln(10,000)
ln V = -kt + ln(10,000)
ln V - ln(10,000) = -kt
ln(V/10,000) = -kt
V = 10,000e^(-kt)
5,000 = 10,000e^(-3k)
0.5 = e^(-3k)
ln(0.5) = -3k
k = ln(2)/3
V = 10,000e^(-2k)
V = 10,000e^(-2ln(2)/3)
V = 10,000 × 2^(-2/3)
V = 10,000 ÷ ∛4
V ≈ $6,300
9️⃣ Common Mistakes to Avoid
❌ Mistake 1: Forgetting +C
Wrong: ∫(1/y) dy = ∫x dx → ln y = x²/2
Right: ∫(1/y) dy = ∫x dx → ln y = x²/2 + C
❌ Mistake 2: Not Separating Properly
Wrong: dy/dx = xy → ∫dy = ∫xy dx
Right: dy/dx = xy → (1/y)dy = x dx → ∫(1/y)dy = ∫x dx
❌ Mistake 3: Sign Errors
"Decreases" means NEGATIVE derivative!
Rate of decrease → dV/dt = -kV (not +kV)
❌ Mistake 4: Forgetting Initial Conditions
Always use given information to find C!
❌ Mistake 5: Not Checking Excluded Values
If you divided by y, check if y = 0 works in the original equation!
🔟 Quick Reference Formulas
📌 Master Formulas
1. Basic Integration Principle∫(dy/dx) dx = y
2. Separation Pattern
If dy/dx = f(x)g(y), then ∫(1/g(y)) dy = ∫f(x) dx
3. Natural Logarithm
∫(1/y) dy = ln|y| + C
4. Exponential Growth
dP/dt = kP → P = P₀e^(kt)
5. Exponential Decay
dP/dt = -kP → P = P₀e^(-kt)
6. Newton's Cooling
dT/dt = -k(T - Tₛ) → T = Tₛ + (T₀ - Tₛ)e^(-kt)
7. From Logs to Exponentials
ln y = f(x) → y = e^(f(x))
8. Combining Constants
e^(A+B) = e^A × e^B
e^(ln C) = C
1️⃣1️⃣ Checklist for Success
✓ Before You Submit Your Answer:
✓ Did you separate variables completely?
✓ Did you include +C when integrating?
✓ Did you apply initial conditions to find C?
✓ Did you solve for the required variable?
✓ Does your answer make sense?
✓ Did you include units if given?
✓ Did you check excluded values?
✓ Did you verify by differentiating?
1️⃣2️⃣ Practice Problems
🏋️ Problem 1: Basic
Solve: dy/dx = 4x³ given y = 2 when x = 1
Click for hint 👆
This is ready to integrate! Just ∫dy = ∫4x³ dx
🏋️ Problem 2: Separation Needed
Solve: dy/dx = x/y given y = 3 when x = 0
Click for hint 👆
Multiply both sides by y to get: y dy = x dx
🏋️ Problem 3: Word Problem
A population of 1000 bacteria doubles every 3 hours. Find the population after 5 hours.
Click for hint 👆
Use dP/dt = kP. Find k using "doubles in 3 hours" means P = 2000 when t = 3
1️⃣3️⃣ Summary
🎓 Key Takeaways
1. What is a Differential Equation?
An equation containing derivatives (dy/dx, dP/dt, etc.)
2. Solution Type
Solutions are functions, not numbers!
3. Main Technique
Separation of Variables: Get f(y) dy = g(x) dx, then integrate
4. General vs Particular
• General: Contains arbitrary constant C
• Particular: Use initial conditions to find C
5. Common Models
• Growth: dP/dt = kP
• Decay: dP/dt = -kP
• Cooling: dT/dt = -k(T - Tₛ)
6. Word Problems
Translate carefully: "rate of decrease" → negative derivative!
💪 Final Tips
- Practice regularly - The more you do, the easier it gets!
- Check your work - Differentiate your answer to verify
- Watch your signs - Increase = +, Decrease = -
- Don't forget +C - It's essential!
- Read carefully - Keywords matter!
- Show all steps - You get marks for method!
🌟 You've Got This! 🌟
Differential equations describe how our world changes.
Master them, and you'll understand growth, decay, motion, and so much more!
📚 Keep practicing and good luck with your exams! 🎯
