HCA Further Mathematics: Differential Equations (2025 syllabus)

HCA Department of Mathematics Differential Equations Further Mathematics 9649 (2025 onward) Gerard Sayson Syllabus requirements ˆ Analytical solution of first order and second order linear differential equations of the form – dy dx + p(x)y = q(x) using an integrating factor – d2y dx2 + a dy dx + by = 0 for a, b∈ R – d2y dx2 + a dy dx + by = f (x) for a, b∈ R where f (x) is a polynomial or pekx or p cos(kx) or q sin(x) including those that can be reduced by means of a given substitution ˆ Relationship between the solution of a non-homogenous equation and the associated homogenous equation ˆ Family of solution curves ˆ Phase lines and slope fields (non-examinable) ˆ Exponential growth model ˆ Logistic growth model, equilibrium points and their stability, and har- vesting Document code: 9649-N-DE15 Class: P Published: 28 November 2024 Revision: N/A 1

Contents 1 Introduction 3 2 First order linear differential equations 5 2.1 Analytical solution for homogenous equations . . . . . . . . . 5 2.2 Analytical solution for non-homogenous equations . . . . . . 5 2.3 Alternate derivation of the solution . . . . . . . . . . . . . . . 7 3 Second order linear differential equations 9 3.1 Analytical solution for homogenous equations . . . . . . . . . 9 3.1.1 Distinct roots . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.2 Repeated roots . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Analytical solution for non-homogenous equations . . . . . . 12 3.2.1 Relationship with homogenous equations . . . . . . . 13 3.2.2 Method of undetermined coefficients . . . . . . . . . . 13 4 F amily of solution curves 15 4.1 Sketching solution curves . . . . . . . . . . . . . . . . . . . . 15 4.2 Phase lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Slope fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Population models 20 5.1 Exponential growth . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Logistic growth . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.3 Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2

1 Introduction “Among all the mathematical disciplines, the theory of differential equations is the most important; it furnishes the explanation of all those elementary manifestations of nature which involve time.” Thus declared Sophus Lie, a famed mathematician responsible for ad- vances in linear and abstract algebra, and most importantly here, differential equations. Differential equations are responsible for virtually every single interac- tion in this universe (this is meant literally): it is the language of Physics and all the other physical sciences. Indeed, these kinds of equations allow us to model ongoing processes that evolve with time, like radioactive decay and population growth. So, what exactly is a differential equation? Differential equations are really just equations where the variable – usually a number in elementary algebra – is replaced by a derivative. It is quite sim