Differential Equations - Formulas & Methods
IB Mathematics Analysis & Approaches (HL)
📐 First Order Differential Equations
General Form:
\[\frac{dy}{dx} = f(x, y)\]
An equation involving an unknown function and its derivative
Solutions:
• General solution: Contains arbitrary constant(s)
• Particular solution: Found using initial/boundary conditions
• Initial condition: Value of y when x = x₀
🔍 Types of First Order Differential Equations
1. Separable: \(\frac{dy}{dx} = g(x)h(y)\) → Separate variables and integrate
2. Homogeneous: \(\frac{dy}{dx} = f\left(\frac{y}{x}\right)\) → Use substitution \(y = vx\)
3. Linear: \(\frac{dy}{dx} + P(x)y = Q(x)\) → Use integrating factor
4. Numerical (Euler's Method): Any form → Iterative approximation
🔀 Separable Differential Equations
Recognizable Form:
\[\frac{dy}{dx} = g(x) \cdot h(y)\]
Function of x multiplied by function of y
Separation Step:
\[\frac{1}{h(y)}\,dy = g(x)\,dx\]
Separate variables: all y terms on left, all x terms on right
Integration Step:
\[\int \frac{1}{h(y)}\,dy = \int g(x)\,dx + C\]
Integrate both sides (one constant of integration needed)
Solution Steps:
1. Check if equation can be written as \(\frac{dy}{dx} = g(x)h(y)\)
2. Rearrange to \(\frac{1}{h(y)}dy = g(x)dx\)
3. Integrate both sides
4. Apply initial conditions to find C
5. Rearrange to solve for y if possible
🔄 Homogeneous Differential Equations
Recognizable Form:
\[\frac{dy}{dx} = f\left(\frac{y}{x}\right)\]
Right-hand side is a function of the ratio \(\frac{y}{x}\) only
Substitution:
\[y = vx\]
\[\frac{dy}{dx} = v + x\frac{dv}{dx}\]
Resulting Equation:
\[v + x\frac{dv}{dx} = f(v)\]
Separable Form:
\[x\frac{dv}{dx} = f(v) - v \quad \Rightarrow \quad \frac{dv}{f(v) - v} = \frac{dx}{x}\]
Solution Steps:
1. Verify equation is in form \(\frac{dy}{dx} = f\left(\frac{y}{x}\right)\)
2. Substitute \(y = vx\) and \(\frac{dy}{dx} = v + x\frac{dv}{dx}\)
3. Simplify to get separable equation in v and x
4. Separate variables and integrate
5. Substitute back \(v = \frac{y}{x}\) to get solution in y and x
⚡ Linear Equations (Integrating Factor)
Standard Form:
\[\frac{dy}{dx} + P(x)y = Q(x)\]
First-order linear differential equation
Integrating Factor:
\[\mu(x) = e^{\int P(x)\,dx}\]
Given in formula booklet
Multiply Through:
\[\mu(x)\frac{dy}{dx} + \mu(x)P(x)y = \frac{d}{dx}[\mu(x)y]\]
Integration:
\[\mu(x)y = \int \mu(x)Q(x)\,dx + C\]
Solution:
\[y = \frac{1}{\mu(x)}\left(\int \mu(x)Q(x)\,dx + C\right)\]
Solution Steps:
1. Write equation in standard form \(\frac{dy}{dx} + P(x)y = Q(x)\)
2. Identify P(x) and Q(x)
3. Calculate integrating factor \(\mu(x) = e^{\int P(x)\,dx}\)
4. Multiply entire equation by \(\mu(x)\)
5. Recognize left side as \(\frac{d}{dx}[\mu(x)y]\)
6. Integrate both sides and solve for y
🔢 Euler's Method (Numerical Solution)
Purpose:
Numerical approximation for differential equations that cannot be solved analytically
Update Formula:
\[x_{n+1} = x_n + h\]
\[y_{n+1} = y_n + h \cdot f(x_n, y_n)\]
where h = step size, f(x, y) = \(\frac{dy}{dx}\)
Algorithm Steps:
1. Start with initial values: \(x_0\), \(y_0\), and step size h
2. Calculate \(f(x_0, y_0)\)
3. Compute \(y_1 = y_0 + h \cdot f(x_0, y_0)\)
4. Compute \(x_1 = x_0 + h\)
5. Repeat for desired number of steps
Important Notes:
• Smaller step size (h) → More accurate approximation
• Accuracy decreases over larger intervals
• Can be implemented on GDC using spreadsheet
• Error accumulates with each step
🎯 Initial & Boundary Conditions
Initial Condition:
• Value of y given when x = x₀ (usually x₀ = 0)
• Written as: y(x₀) = y₀ or "y = y₀ when x = x₀"
• Used to find the constant of integration C
• Converts general solution to particular solution
Finding the Constant:
1. Solve the differential equation to get general solution
2. Substitute the initial condition values
3. Solve for the constant C
4. Write the particular solution with the value of C
Common Phrases:
• "Initially" or "at the start" → t = 0
• "When x = a, y = b" → boundary condition
• "Passes through the point (a, b)" → y = b when x = a
• "Starts at the origin" → y = 0 when x = 0
🌍 Modeling Applications
Exponential Growth/Decay:
\[\frac{dy}{dt} = ky\]
Solution: \(y = Ae^{kt}\) (k > 0 for growth, k < 0 for decay)
Newton's Law of Cooling:
\[\frac{dT}{dt} = -k(T - T_s)\]
where T = temperature, \(T_s\) = surrounding temperature
Logistic Growth:
\[\frac{dP}{dt} = kP(L - P)\]
where P = population, L = carrying capacity (maximum)
Common Real-World Models:
• Population growth
• Radioactive decay
• Chemical reactions
• Temperature changes (cooling/heating)
• Investment growth
• Epidemiology (disease spread)
✅ Problem-Solving Strategy
Identifying the Type:
• Can variables be separated? → Separable
• Is it in form \(\frac{dy}{dx} = f\left(\frac{y}{x}\right)\)? → Homogeneous
• Is it in form \(\frac{dy}{dx} + P(x)y = Q(x)\)? → Use integrating factor
• None of the above? → Use Euler's method for approximation
General Approach:
1. Identify the type of differential equation
2. Choose the appropriate method
3. Apply the method systematically
4. Don't forget integration constants
5. Apply initial/boundary conditions
6. Write the particular solution
7. Check your answer by differentiating
Common Techniques Needed:
• Partial fractions for complex integrals
• Integration by substitution
• Integration by parts
• Logarithmic and exponential manipulation
• Algebraic rearrangement
💡 Exam Tip: Differential equations are HL only! The integrating factor formula is given in the formula booklet. Always start by identifying the type - this determines your method. For separable equations, check if partial fractions are needed. For homogeneous equations, remember the substitution y = vx. For linear equations, don't forget to rearrange to standard form first. Euler's method is great for numerical approximation - can be done efficiently on GDC using spreadsheet mode. Practice recognizing which method to use!