# First-Order Differential Equations and Euler’s Method (HL) | Maths with Lemon

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### Separable & Homogeneous Differential Equations

IB Mathematics: Analysis and Approaches Higher Level · AHL 5.18 · Lesson 10.3. Solve first-order differential equations using separation of variables and the substitution \(y=vx\), and model growth and decay.

Lesson 10.3 worksheet (PDF) Previous lesson: Euler’s Method Calculus overview


#### 1. Separation of Variables


##### What you need to know

- A separable equation can be written as a product of a function of \(x\) and a function of \(y\).
- Move the \(y\)-terms beside \(dy\), and the \(x\)-terms beside \(dx\).
- Integrate both sides, including a constant.
- Check constant solutions before dividing by \(g(y)\).

##### Essential formula


#### 2. Worked Example: Separable Equation


##### Method

- Separate the variables.
- Integrate both sides.
- Rearrange the result.
- Apply initial conditions, if given.

##### From the worksheet

Separate:

Integrate:

Exponentiate:


#### 3. Particular Solutions and Initial Conditions


##### Remember

- A general solution includes an arbitrary constant.
- An initial condition determines the constant.
- Pay attention to the correct square-root branch and domain.
- An implicit answer can sometimes be acceptable.

##### Worksheet example

Use \(y(0)=10\), giving \(C=51\).

The positive branch is required by the initial condition.


#### 4. Homogeneous Differential Equations


##### Recognising the structure

- The right-hand side is a function of the ratio \(y/x\).
- Set \(y=vx\), with \(v\) a function of \(x\).
- Use the product rule when differentiating \(vx\).
- The new equation is separable.

##### Substitution


#### 5. Worked Example: Homogeneous Equation


##### Four-step process

- Substitute \(y=vx\).
- Simplify to find \(dv/dx\).
- Separate and integrate.
- Replace \(v\) with \(y/x\).

##### From the worksheet

Let \(y=vx\):

Replace \(v=y/x\):


#### 6. Growth, Decay and Logistic Equations


##### Applications

- Radioactive decay: rate proportional to mass remaining.
- Bacterial growth: rate proportional to population size.
- Logistic growth: growth slows near carrying capacity.
- The logistic equation can require partial fractions.

##### Key models

Initially 25 g, decreasing at 5 g/s. For \(dm/dt=-km\), we have \(k=1/5\).

Half-life:


#### 7. Interactive Growth and Decay Explorer


##### Explore the model

- Choose growth or decay.
- Change the initial value and rate constant.
- Observe how the solution changes over time.
- Compare the exact exponential solution with your calculations.

##### Exact solution of \(dN/dt=kN\)

This graph uses the exact exponential solution, not Euler’s approximation.


#### 8. HL Practice and Revision


##### Practice from Lesson 10.3

- Separable equations such as \(dy/dx=x^3y^3\).
- Homogeneous equations such as \(dy/dx=(x^2-y^2)/(xy)\).
- Particular solutions with initial conditions.
- Growth, decay and substitution challenges.

##### Common mistakes

Dividing by a function of \(y\) can discard solutions where that function is zero.

For integrals arising from equations such as the logistic model.

Open the complete worksheet


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