# FP3 Tutorials – OCR A Level Maths

## Contents

## First Order Linear Differential Equations

## Separating the Variables

## Exact Equations (Integrating Factors)

## Substitution Types

## Second Order Linear Differential Equations

## Equations of the form a d²y/dx² + b dy/dx + cy = 0

## Equations of the form a d²y/dx² + b dy/dx + cy = f(x)

- General solutions where f(x) = k (constant types)
- General solutions where f(x) = kx (linear types)
- General solutions where f(x) = kx
^{2}(quadratic types) - General solutions where f(x) = ke
^{px}(exponential types) - General solutions where f(x) = λ cosωx + µ sinωx (trig types)
- Special types of particular integrals
- Exam Questions - General solutions where f(x) = kx (linear types)
- Particular solutions using boundary conditions to solve differential equations
- Exam Questions - Exponential Type ke
^{px}(exponential types) - Exam Questions - Trig Type
- Exam Questions - Particular solutions using boundary conditions

## Using a Substitution

## Groups

## Groups

- What defines a group
- Different types of groups
- Commutative groups
- Order of a group
- Subgroups
- Lagrange’s theorem
- Cyclic groups
- Generators
- Proof that all cyclic groups are Abelian
- Permutations under composition of permutations
- Cycle notation for permutations
- Result that every permutation can be written as a composition of disjoint cycles
- The order of a combination of cycles