Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Differential Equations

Solve separable, homogeneous and linear first-order equations and approximate solutions with Euler’s method.

AA HL · AHL 5.18

Learning goal

Use the four AA HL methods: separation of variables, y=vx for homogeneous equations, integrating factors, and Euler’s method.

Syllabus link

AA HL: AHL 5.18 · Topic 5 Calculus.

Big idea

A differential equation specifies how a quantity changes. Analytic methods produce families/particular solutions; Euler’s method builds a numerical path from a starting point.

Key relationship

\(\frac{dy}{dx}=f(x)g(y)\Rightarrow\frac{1}{g(y)}dy=f(x)dx\)

1

Core idea

A differential equation specifies how a quantity changes. Analytic methods produce families/particular solutions; Euler’s method builds a numerical path from a starting point.

Use the four AA HL methods: separation of variables, y=vx for homogeneous equations, integrating factors, and Euler’s method.

2

Key results and method

\[\frac{dy}{dx}=f(x)g(y)\Rightarrow\frac{1}{g(y)}dy=f(x)dx\]
\[y=vx\Rightarrow\frac{dy}{dx}=v+x\frac{dv}{dx}\]
\[y\'+P(x)y=Q(x),\;\mu=e^{\int P(x)dx}\]
\[y_{n+1}=y_n+h f(x_n,y_n)\]
  1. Identify the equation type before manipulating it.
  2. Use initial conditions only after obtaining the general relation/solution unless the numerical method requires the starting point.
  3. For Euler, advance \(x\) by fixed step \(h\) and use the current slope to update \(y\).
3

Worked example

Question

Solve \(dy/dx=xy\) with \(y(0)=2\).

Solution
  1. Separate: \(dy/y=x\,dx\).
  2. Integrate: \(\ln|y|=x^2/2+C\).
  3. Exponentiate and use \(y(0)=2\).
Answer: \(y=2e^{x^2/2}\).
4

Practice

Try these questions before moving on.

  1. Solve a logistic differential equation and interpret equilibria/carrying capacity.
  2. Solve a homogeneous equation using \(y=vx\).
  3. Solve a linear equation with an integrating factor.
  4. Perform two Euler steps from a stated initial condition.

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ Euler’s method on TI-Nspire CX ▶ Homogeneous differential equations ▶ Maclaurin-series differential equation example
Premium printable lesson notes

The finalized printable handout for this topic is a premium Radford Mathematics resource and is not offered as a free website download.

Visit the Radford Mathematics Store →