Topic : Applications of Differential Equations S3S

Q1. An object is released from rest . Apart from being acted on by the gravitational pull g , the resistance that the object encounters varies with its velocity v m/s , thus giving its acceleration a as g-kv , where k

is positive constant . Show that . (i) By considering t , find the velocity it obtained . (ii) Sketch the graph of against t .

Q2. The number of organisms in a population at time t is denoted by x . Treating x as a continuous variable , the

differential equation is satisfied by x and t is , where k is a positive constant .

(i) Given that x = 10 when t=10 ,solve the deferential equation , obtaining a relation between x , k and t .

(ii) Given also that x = 20 when t = 1 , show that k = .

(iii) Show that the number of organisms never reaches 48 , however large t becomes . [x= ; 47.833]

Q3. An infectious disease spreads at the rate which is proportional to the product of the number infected and the number uninfected . Initially , one half of the population is infected and the rate of spread is such that , were it to remain constant , the whole population would become infected in 24 days . Calculate the proportion of the population which is infected after 12 days . [ans : 73 % ]

Q4. Suppose that a chemical mixture contains two substances whose masses are x kg and y kg , and whose combined mass is 1 kg . At any time t , the rate at which x is increasing is proportional to the product of the two masses at that time . Obtain a differential equation relating x and t . Show that the general solution of the differential

equation maybe expressed in the form , where C and k are positive constants . Given that initially

x = 1/10 and when t=1 , x = 1/4 , express x in terms of t . Show that x → as . [ans: x = ]

Q5. Experiments show that the rate of cooling of a metallic object is proportional to the difference between its temperature T(t) and that of its environment , T env (which is a constant ) . (a) Write down a differential equation describing this situation . (b) The object is heated to 1200 C and then left to cool in a large room at 250 C . Its temperature falls to 780C in 30 minutes . Find its temperature after 3 hours of cooling . [ans : 27.90 ]

Q6. In a college hostel accommodating 1000 students , one of them came in carrying a flu virus , then the hostel was isolated . If the rate at which the virus spreads is assumed to be proportional not only to the number N of infected students , but also to the number of non-infected students , and if the number of infected students is 50 after 4 Z days , then show that more than 95% of the students will be infected after 10 days .

Q7. The population of a village increases at the rate proportional to the number of its inhabitants present at any time . If the population of the village was 20,000 in 2010 and 25,000 in the year 2015 , what will be the population of the village in 2020 ?

Q8. A corpse is discovered in a hotel room . At midnight . a police detective found the body’s temperature to be 300C . At 2 a.m. a medical examiner measures the body’s temperature to be 240C . Assuming the room in which the body was found had a constant temperature of 21o C , estimate the time of death . (Recall that the normal human body temperature is 370 C ) (ans : 10:57 p.m )

Q9. An electric circuit contains a power source with time dependent voltage of E(t) volts , a resistor with a constant resistance R ohms , and an inductor with a constant inductance of L henrys . The current I(t) amperes flows

through the circuit is given by the equation . Find the expression I as a function of t when E = 0 . Hence , show that the value of the current when t = L/R is 1/e . (L/R is called time constant of the RL circuit )

Q9. Hydrocodone bitartrate is used as a cough suppressant . After the drug is fully absorbed , the quantity of drug in the body decreases at a rate proportional to the amount left in the body , The half-life of hydrocodone bitartrate in the body is 3.8 hours , and the usual oral dose is 10mg . (a) Write a differential equation for the quantity , Q , of hydrocodone bitartrate in the body at a time t , in hours , since the drug was fully absorbed . (b) Solve the differential equation . (c) Find how much of the 10mg dose is still in the body after 12 hours ? [ans : 1.126mg]

Q10. An anthropologist is modelling the population of the island of A . In the model , the population at the start of the year t is P . The birth rate is 10 births per 1000 population per year . The death rate is m deaths per 1000 population per year .

(a) Show that

(b) At the start of year 0 the population was 108000 . Find an expression for P in terms of t . (c) If the population is to double in 100 years , find the value of m .

[ans : ; m = 10(1-ln2)]