ImperialX: A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices

How to find the general or particular solution to a first-order differential equation by inspection or by using an integrating factor.

How to find a numerical solution to a differential equation using the Euler method or an improved Euler method..

How to find the nth roots of unity

How to find the nth roots of a complex number in the form

How to use complex roots of unity to solve geometrical problems.

How to identify the features of parabolas, rectangular hyperbolae, ellipses and hyperbolae defined by Cartesian and parametric equations.

How to identify features of graphs defined by rational functions.

How to define a parabola, ellipse or hyperbola using focus-directrix properties and eccentricity.

How to evaluate improper integrals.

How to integrate using partial fractions

How to derive and use reduction formulae

How to find areas enclosed by curves that are defined parametrically.

How to find the area enclosed by a polar curve.

How to calculate arc length.

How to calculate the surface area of revolution.

How to find the auxiliary equation for a second order differential equation.

Module 1: First Order Differential Equations

• Solving first order differential equations by inspection
• Solving first order differential equations using an integrating factor
• Finding general and particular solutions of first-order differential equations
• Euler’s method for finding the numerical solution of a differential equation
• Improved Euler methods for solving differential equations.

Module 2: Further Complex Numbers

• The nth roots of unity and their geometrical representation
• The nth roots of a complex number and their geometrical representation
• Solving geometrical problems using complex numbers.

Module 3: Properties of Curves

• Cartesian and parametric equations for the parabola and rectangular hyperbola, ellipse and hyperbola.
• Graphs of rational functions
• Graphs of , , for given
• The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity.

Module 4: Further Integration Methods

• Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.
• Integrate using partial fractions including those with quadratic factors in the denominator
• Selecting the correct substitution to integrate by substitution.
• Deriving and using reduction formula

Module 5: Further Applications of Integration

• Finding areas enclosed by curves that are defined parametrically
• Finding the area enclosed by a polar curve
• Using integration methods to calculate the arc length
• Using integration methods to calculate the surface area of revolution

Module 6: Second Order Differential Equations

• Solving differential equations of form y″ + ay′ + by = 0 where a and b are constants by using the auxiliary equation.
• Solving differential equations of form y ″+ a y ′+ b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function

Module 7: The Vector (cross) Product

• The definition and properties of the vector product
• Using the vector product to calculate areas of triangles.
• The vector triple product.
• Using the vector triple product to calculate the volume of a tetrahedron and the volume of a parallelepiped
• The vector product form of the vector equation of a straight line
• Solving geometrical problems using the vector product

Module 8: Matrices - Eigenvalues and Eigenvectors

• Calculating eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices.
• Reducing matrices to diagonal form.
• Using the Cayley-Hamilton Theorem

Is this course for a particular examination board?
This course covers content that can be found in all of the A level Further Mathematics specifications. Some of the content covered is considered to be at AS level for some exam boards whilst being at A level for others.