ACJC 2026 Differential Equations Lecture Notes

12 DIFFERENTIAL EQUATIONS SYLLABUS • Solving for the general solutions and particular solutions of differential equations of the form d f ( )g( )d y xyx = - Use direct integration to solve differential equations of the form d f ( )d y xx = ; - Use the method of separable variables to solve differential equations of the form d g( )d y yx = and d f ( )g( )d y xyx = ; - Use a given substitution to reduce a first -order differential equation to the form d f ( )g( )d y xyx = . • Formulating a differential equation from a problem situation - Formulate a simple statement involving a rate of change as a differential equation, including the introduction of a constant of proportionality from a problem situation if necessary. • Interpreting a differential equation and its solution in terms of a problem situation - Comment on the appropriateness of the models used and the assumptions made.

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Introduction ..................................................................................... 3 2 First Order Differential Equations .................................................... 5 2.1 Differential Equations of the Form d f ( )d y xx = ........................ 5 2.2 Differential Equations of the Form d g( )d y yx = ....................... 6 2.3 Differential Equations of the Form d f ( )g( )d y xyx = ................ 8 3 Reduction of a Differential Equation by Means of a Suitable Substitution ..................................................................................... 10 4 Formulating and Solving a Differential Equation from a Problem Situation .......................................................................................... 13 5 More Worked Examples ................................................................. 19 Annex: Practice Questions on Differential Equations ....................... 23

12 Differential Equations 3 LECTURE 1 Lesson Outline • Introduction to ordinary differential equations. • Solving for the general solutions and particular solutions of first order differential equations of the form d f ( )g( )d y xyx = . 1 INTRODUCTION A differential equation (DE) is an equation that contains one or more functions together with their derivatives. DEs allow modelling of laws governing natural phenomena which involve quantities and their rates of change. Researchers solve the DEs to analyse the relationship between the variables and predict future behaviour. For this reason, DEs are used in physics, chemistry, biology and economics. Example A researcher observes that the rate of increase of the population of a particular species of bird is proportional to the number of birds, x, present at time t weeks. This situation can be modelled by a differential equation in x, d d x kxt = , where k is a positive constant. Suppose k = 0.01. This means an increase (i.e. growth rate) of 0.01 birds per week for every existing bird at that instant. At the instant when 1000x= , i.e. on the week that the number of birds is 1000, the rate of increase of the population is 0.01  1000 = 10 new birds. At the instant when 2000x= , the rate of increase of population is 0. 01  2000 = 20 new birds. So, the rate of increase of population (at any instant) is the growth rate multiplied by the population at that instant. Notice that the population is constantly increasing: the more birds we have, the more new baby birds we get. The researcher can solve this differential equation, i.e. find x as a function of t, to predict the future population using this model.

ACJC 2025/26 H2 Mathematics (9758) 4 DEFINITIONS 1. An ordinary differential equation is an equation relating x, y and the derivatives d d y x , 2 2 d d y x , and so on. For example, 2 d 2d yy xy xx+= and 2 2 dd 3 4 sindd yy yxxx − + = . 2. The order of a differential equation is the order of the highest derivative that occurs in it. 3. The degree of a differential equation is the power of the highest derivative that occurs in it. 4. The general solution of a differential equation of order n is a function that satisfies the differential equation and it contains n arbitrary constants. Example Solve the differential equation d 2 . (1)d y xx = −−− 2 22 d 2 2 xy x x C x C = = + = + The general solution of the differential equation (of order 1) is 2y x C=+ . (It has one arbitrary constant, C.) 5. A particular solution of a differential equation is a solution of the differential equation which contains no arbitrary constants. Example 2 1yx=+ , 2 1yx=− and 2 2yx=+ are some of the p articular solutions of equation (1). A particular solution of a differential equation can be found if initial conditions are given. Differential Equation Order Degree 2 d 20d yxy x+ − = 1 1 2 dd sindd yy yxxx  + − = 1 2 4 52 22 2 dd edd xyy xxx  +=  2 4

12 Differential Equations 5 Example Find the particular solution of d 2d y xx = given that y = 7 when x = 2. From (1), general solution is 2y x C=+ . When x = 2, y = 7, 27 2 3 CC= +  =  The particular solution is 2 3yx=+ . 2 FIRST ORDER DIFFERENTIAL EQUATIONS 2.1 Differential Equations of the Form =d f( )d y xx A differential equation of the form d f ( )d y xx = can be solved by directly integrating the equation with respect to x. For example, given the differential equation 2d d y xx = , we integrate both sides w.r.t. x: 23 1d 3y x x x C= = + . Example 1 Solve the following differential equations: (a) 2d 12d yxx x−= (b) 2d lnd w sss = Solution (a) Rearrange 2d 12d yxx x−= into the form d f ( )d y xx = : 2d 2 1 d yx xx += Integrate both sides w.r.t. x, 2 2 d 2 1ddd 11 d 2 d ln yx xxxx y x x x y x x C += =+ = + +   (b) 2d lnd w sss = Integrate both sides w.r.t. s, 2d d ln dd w s s s ss = This is an initial condition to find the specific value of C. This is the general solution for the given DE. Must use modulus for ln. RHS can be solved using integration by parts.

ACJC 2025/26 H2 Mathematics (9758) 6 33 3 3 3 2 1ln d33 1ln d ln3 3 3 9 = −  = − = − +   ssw s s s s s sw s s s s C 2.2 Differential Equations of the Form =d g( )d y yx We cannot integrate both sides with respect to x directly. We instead rearrange the differential equation d g( )d y yx = into the form 1d 1g( ) d y yx = . Integrate both sides with respect to x, 1 d 1 d 1 d d 1 dg( ) d g( ) y x x y xy x y =  =    We effectively separated the variables of the differential equation, putting the term containing y on the side with d d y x . For example, given the differential equation d 2d y yx = . we rearrange the equation such that the variables are separated: 1 d 2 1 dyxy = 2 2 2 2 ln 2 e e ee e , e xC xC xC xC y x C y y y y A A + + =+ = = = = = Example 2 (a) Find the general solution of the differential equation 2d 12d yyy x−= . (b) The gradient of a curve at the point ( , )xy is given by 2 2 1 y y + . Find the equation of the curve with this gradient function which passes through the origin. Solution (a) 2d 12d yyy x−= 2d 2 1 d +=yy xy The variables are separated with y on LHS and 1 on RHS. A is an arbitrary constant (independent of x). ■

12 Differential Equations 7 ( ) 2 2 2 2 d 12 1 d d 1 d21 14 d4 2 1 1 ln 2 14 =+ =+ =++ + = +   yy yx y yxy y y x Cy y x C We can stop here and leave the answer as above since the question did not require us to “express y in terms of x”. However, if we are required to express y in terms of x for (a), the next few steps are required: ( ) 2 2 4 4 44 2 44 2 4 24 4 ln 2 1 4 4 2 1 e e1 2 e e 1 2 e1 , where e 2 e1 , where is an arbitrary constant.2 xC xC xC x C x y x C y y y AyA AyA + + + = + += −= −= −== −= (b) 2