RVHS H2 Math Differential equations Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differential Equations 1 Differential Equations 1. AJC/2009/II/3 (a) By using the substitution z x y=+ , show that the differential equation ( ) 2d 1d yx y x xy xx+ = + + + can be reduced to the equation ( )( )d 11d zz x zx = + + . Hence find the general solution of the differential equation ( ) 2d 1d yx y x xy xx+ = + + + . [4] (b) John jogs 8 km every morning. As a mathematical model, he assumes that his speed during the jog is proportional to the distance he has yet to complete. (i) He starts off at an initial speed of 10 km per hour, and after t hours, he has jogged x km. Show that ( )d5 8d4 x xt =− . [1] (ii) Find an expression for x in terms of t and hence find the time taken for John to complete 6 km. [3] (iii) Comment on the suitability of the model. [1] 2. EJC Prelim/2020/01/Q9 (a) (i) Given x and y are related by the differential equation 2 d d yx xy kx+= for 𝑘 ∈ ℝ, show that ( )lnkxy x += is a solution of the differential equation where  is an arbitrary constant. [2] (ii) Hence, show that ( ) 11e , e k−− is a stationary point of the curve ( )lnkxy x += . [2] (b) It is given that x and y are related by the differential equation 22d d yy x x yx+ = + and that 0y= when 2x=− . (i) By substituting 22v x y=+ , show that the differential equation can be written as d 2d v vx = . [2] (ii) Find v in terms of x and hence show that ( ) 2 fyx= where ( )f x is to be determined. [4]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differential Equations 2 3. NYJC/II/2 In a controlled experiment, the rate at which liquid fuel is pumped into a tank is twice the remaining volume of the tank, and the rate at which the fuel in the tank is used up is proportional to the square of the volume of fuel in the tank. The capacity of the tank is 33 m2 and the volume of fuel in the tank at time t is 3 mx . (i) Given that the volume of fuel in the tank stabilizes at 31 m , show that the increase of the fuel is given by the differential equation ( ) 2d 23d x xxt =− + − . (ii) Given that the tank is initially empty, find x in terms of t. (iii) Sketch the graph of your solution curve in (ii). 4. SAJC/II/1 The population size of a colony of ants increases at a rate which is inversely proportional to the population size and decreases at a rate which is proportional to the population size. The population size of the colony (in thousands) is x at time t days, where 2x . Given that when the population size is 2000, it remains at this value, show that d𝑥 d𝑡 = 𝑘 ( 𝑥2−4 𝑥 ) where k is a negative constant. Hence find the general solution of the differential equation by expressing x in terms of t. [5] 5. YJC/1/6 In a chemical reaction, the amount of substance X and substance Y are being observed. There are x grams of substance X and y grams of substance Y at time t minutes. Using a mathematical model, the variable x and y satisfy the equations xt y −= 2d d and xet x =d d , for 10  t . (i) Initially, there are no traces of substance X and substance Y present in the reaction. Express y in terms of x. [4] (ii) The amount of substance Y during the reaction is the maximum at time k minutes. Find the value of k. [4]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differential Equations 3 6. NYJC/2010/I/10 An innovation is introduced into a community of 100 farmers at time 0t = . Let x denote the number of farmers who have adopted the innovation at time t. Assume that x is a continuous function of time. The rate at which the number of farmers in that community who adopted the innovation at a particular instant is proportional to the product of the number of farmers who have already adopted and the number of farmers who have not adopted the innovation. Initially, one farmer adopted the innovation and the rate at which the number of farmers who adopted the innovation is one farmer per unit time. (i) Show that dx dt = k(100x− x2) , where k is to be determined. [1] (ii) Find the particular solution of x , in terms of t. [5] (iii) Sketch the graph of x versus t, for 0t . [2] (iv) Using the graph in (iii) or otherwise, find the time taken for 75% of the population of farmers to adopt the innovation, leaving your answer to 2 decimal places. [1] (v) Give a reason why the model may not be suitable. [1] 7. NYJC/2011/I/6 (a) The variables y and x are related by 2d 2d yy xxx=− . By using the substitution y = ux, solve this equation and hence find y in terms of x, given that y = 0 when x = 1. (b) In a pathological investigation of the spread of disease in a country, it is found that the rate at which the proportion of area infected at any time (hours) is proportional to the product of the proportion of area infected and the proportion of area not infected. Initially, half of the area of the country is infected. When proportion of area infected increases at a rate of 1 54 per hour, the proportion of area infected is 1 3 . The proportion of area infected at any time t is given by x. (i) Obtain a differential equation relating x and t and show that the particular solution of the differential equation is of the form 121 t Ax Be − = + , where A and B are positive constants to be determined. (ii) Find the proportion of the area infected after 12 hours. 8. DHS/2012/I/11 Given that 2 de d x yz x= , express d d z x in terms of x, d d y x and 2 2 d d y x . Hence show that the differential equation 2 14 2 dd 2edd xyy xx −+= can be reduced to 12d ed xz x −= . Solve this differential equation, expressing y in terms of x. [5]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differential Equations 4 9. MJC/2012/I/10 (a) By means of the substitution y xz= , show that the differential equation ( ) ( ) 2de 1 e 1 d xx y y x x x y + − = − can be reduced to the form d e 1 d e 1 x x zz x −= + . Hence find the general solution for 2y in terms of x. [5] (b) There was an island where initially there was no one living on it. The total capacity of the island is 9 000. The population increases at a rate which is inversely proportional to the remaining capacity of the island. At the same time, the rate at which the population decreases is 1 20 of the population size. When the population reaches 4 000, it remains at this value. The population size (in thousands) is x at time t months, show that ( )( ) ( ) 45d .d 20 9 xxx tx −−= − Find t in terms of x. Hence find the time when the population reaches 2 000. [7] 10. VJC Prelim 9758/2023/01/Q11 Statistics show that the population of rodents in a small area follows a density-dependent population growth model. In this model, as the population approaches its carrying capacity, the growth rate slows down due to increased competition for limited resources, such as food, space, and mates. In a particular small farm, a virus spread has a significant impact on the population of rodents. The population t days after the start of the virus spread is denoted by N (in hundreds). The density- dependent population growth model states that the population of rodents increases at a rate inversely proportional to the square of the population. At the same time rodents die due to the virus spread at a rate proportional to the population. When N = 3, the number of rodents remains constant. (a) Show that 3 2 d (27 ) d N k N tN −= , where k i