DEFINITION DIFFERENTIAL EQUATIONS SEPARATING THE VARIABLES. A differential equation is an equation dy. involving a differential coefficient i.e.

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1 FIRST ORDER DEFINITION A differential equation is an equation involving a differential coefficient i.e. In this sllabus, we will onl learn the first order To solve differential equation, we integrate and find the equation which satisfies the differential equation Eample General Solution Particular Solution + c Eample General Solution Particular Solution + c + c +7 Eample General Solution Particular Solution ( ) ( ) ( ) ( ) + c + c ( ) + 0 SEPARATING THE VARIABLES HOW DO WE KNOW IF IT CAN BE SEPARATED? ( )

2 To put it simpl, ou separate the variables of and on the left on the right To put it simpl, ou separate the variables of and on the left ( + ) ( + ) on the right ( + ) ( + ) ( + ) 7 8 To put it simpl, ou separate the variables of and on the left ( ) 5 ( ) 5 on the right CANNOT BE SEPARATED SOLVING FIRST ORDER 9 0 METHOD Separate the variables and Integrate the left hand side in terms of and the right hand side in terms of f() g() f() g() Eample : Find the general solution of the differential.(a) ln + c

3 Eample : Find the general solution of the differential.(b) Eample : Find the general solution of the differential.(c) cosec 5 cosec sin 5 5 sin sin ln 5 + c cos + c Eample : Find the general solution of the differential.(d) ( + ) Eample : Find the general solution of the differential ( + ) 0 ( + ).(e) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ln + c ln( + ) ln( + ) + c ln + c 5 Eample : Solve the following differential :.(a) c + + c Eample : Solve the following differential :.(b), c + +c +c 5 5, 0 ln 5 + c ln 5(0) + c c ln ln 5 + c c ln 5 + ln c e5+ln + + e5 eln e5 7 8

4 Eample : Solve the following differential :.(c) ln + c ln + c c ln + c, 0 sin + c cos sin cos sin cos sin +c ln sin sin sin 0 + c ln + c.(d), Eample : Solve the following differential : sin + c ln (0) sin (0) + c 0 0+c c sin Eample : Solve the following differential :.(e) ( + ), ( + ) ln ln( + ) + c ln ln( + ) + c ( + ) ln ln( + ) + c 0 ln + c c ln APPLICATION ON ln ln( + ) ln APPLICATION ON APPLICATION ON Rate of change of a quantit Q is Eamples are real life proportional to the value Q αq Natural Growth Rate of change is taken with respect to time Natural Deca Newton s Law of Cooling Rate of Liquid Leaking

5 APPLICATION ON NATURAL GROWTH AND NATURAL DECAY () Natural Growth If Q is increasing with time, it can be epressed as differential equation α Q kq Solve the first order differential equation: Q to the left... t to the right k Q Q k ln Q kt + c Q e kt+c Q e kt e c Q Ae kt Rate of increase of number of cells of east is proportional to the number of cells 5 6 (II) Natural Deca If the rate of decrease of Q is proportional to Q, α Q Solve the first order differential equation: Q to the left... t to the right k Q Q k ln Q kt + c kq Q e kt+c Q e kt e c Q Ae kt METHOD Interpret the word problem into a differential equation Solve the first order of differential equation Find the value of k and A (or c) Answer the question 7 8 Eample (): The rate of increase of a colon of bacteria is proportional to the number of bacteria present at an particular time. If initial number of bacteria is 00 and hrs later there are 50, then find k and A. Eample (): The rate of increase of a colon of bacteria is proportional to the number of bacteria present at an particular time. If initial number of bacteria is 00 and hrs later there are 50, then find k and A. Let bacteria be B. db α B db kb db k B B db k ln B kt + c B e kt+c B Ae kt k is positive because it is a rate of increase B Ae kt At t 0, B 00 At t, B Ae k0 00 Ae 0 A 00 k and A? 50 Ae k 50 00e k ek e k.5 k ln(.5) k ln(.5) k

6 Eample (): The rate of increase of a population of tigers is proportional to the population at an particular instant of time. If initial number of tigers is 00 and if after ears there are 80, how man tigers will there be after 5 ears? Eample (): The rate of increase of a population of tigers is proportional to the population at an particular instant of time. If initial number of tigers is 00 and if after ears there are 80, how man tigers will there be after 5 ears? Q Aekt Let tigers be Q. α Q At t 0, Q 00 At t, Q 80 kq k is positive because it is a rate of increase Q 00e ln(.) t 00 Aek0 k Q k Q Q 00e 00 Ae0 80 Ae k Q ekt+c k 80 ek 00 kt Q 6 tigers k ln(.) 80 00ek ln(.) (5) Q A 00 ln Q kt + c Q Ae At t 5, Q? ln(.) ek. Eample (): The number of insects in a population t das after the start of observations is denoted b N. The variation in the number of insects is modelled b a differential equation of the form dn kn cos(t) Eample (): (ii)given also that N 66 when t 0, find the value of k. N 5ek where k is a constant and N is taken to be a continuous variable. It is given that N 5 where t 0. N Ae dn k cos(t) N dn k cos(t) N ln N k 66 5ek 5 Ae sin(t) +c N ek sin(t) sin((0)) ln.8 k 5 Ae0 A 5 k N 5e k sin(0.6) sin(0.6) 66 ek 5.8 ek k sin((0)) 66 5ek sin(t) At t 0, N 5 sin(t) +c N ek k sin(t) At t 0, N 66 (i) Solve the differential equation, obtaining a relation between N, k and t. dn kn cos(t) sin(t) sin(0.6) sin(0.6) calculator in radians! (ln.8)() sin 0.6 k ec k Eample (): (iii)obtain an epression for N in terms of t, and find the least value of N predicted b this model. N 5e0.000 sin(t) k APPLICATION ON k sin t sin(t) (0.000) sin(t) (0.000) (0.000) sin(t) (0.000) sin(t) 0.50 N 5e NEWTON S LAW OF COOLING 0.50 N N

7 Eample (): A cup of coffee, originall at 90 C is left to cool in a room at a constant 0 C. After 5 minutes, the temperature is 80 C. Find how long the coffee takes to cool to 60 C. NEWTON S LAW OF COOLING The rate of cooling of a bo is proportional to the difference between it s temperature and the temperature of the surroundings At t 0 Initial Bo temperature Let θ be the temperature of the bo Let θ0 be the temperature of the surroundings α (θ θ0 ) ln(θ θ0 ) kt + ln C At t 5, θ 80 C Surrounding temperature 0 C k(θ θ0 ) k(θ θ0 ) α (θ θ0 ) ln(80 0) k(5) + ln 70 ln 60 5k + ln 70 k θ θ0 k θ θ0 k θ θ0 k θ θ0 5k ln 60 ln 70 k k ln(θ θ0 ) kt + c ln(θ θ0 ) kt + c ln 60 ln 70 5 ln(90 0) k(0) + c ln(θ θ0 ) 0.008t + ln 70 c ln 70 7 Eample (): A cup of coffee, originall at 90 C is left to cool in a room at a constant 0 C. After 5 minutes, the temperature is 80 C. Find how long the coffee takes to cool to 60 C. 8 Eample (): A liquid is heated in an oven kept at a constant 80 C. It is assumed that the rate of increase in the temperature of the liquid is proportional to (80 C θ) where θ is the temperature of the liquid at time t minutes. If the temperature rises from 0 C to 0 C in 5 minutes, find the temperature of the liquid after a further 5 minutes. ln(θ θ0 ) 0.008t + ln 70 ln(80 0) k(0) + c At t 0 Initial Bo temperature 0 C At t?, θ 60 C c ln(80) Surrounding temperature 80 C ln(80 θ) kt ln 80 Surrounding temperature 0 C ln(60 0) 0.008t + ln 70 α (80 C θ) At t 5, θ 0 k(80 θ) ln t + ln t ln 70 ln 0 ln(80 0) k(5) ln 80 ln 60 5k ln 80 k (80 θ) k (80 θ) ln 70 ln 0 t t 8. 5k ln 80 ln 60 ln 80 ln 60 5 k 0.0 k ln(80 θ) kt + c ln(80 θ) 0.0t ln 80 9 Eample (): A liquid is heated in an oven kept at a constant 80 C. It is assumed that the rate of increase in the temperature of the liquid is proportional to (80 C θ) where θ is the temperature of the liquid at time t minutes. If the temperature rises from 0 C to 0 C in 5 minutes, find the temperature of the liquid after a further 5 minutes. Further 5 minutes t 0 θ? 0 Eample (): The temperature of a quantit of liquid at time t is θ. The liquid is cooling in an atmosphere whose temperature is constant and equal to A. The rate of decrease of θ is proportional to the temperature difference (θ A). Thus θ and t satisf the differential equation. k(θ A) where k is a constant. (i) Find, in an form, the solution to this differential equation, given that θ A when t 0 ln(80 θ) 0.0t ln 80 k(θ A) ln(80 θ) 0.0(0) ln 80 ln(80 θ) 0.0(0) + ln θ e 0.0(0)+ln 80 θ 80 e 0.0(0)+ln 80 k (θ A) k (θ A) ln(θ A) kt + c θ 80 0 ln(a A) k(0) + c c ln A θ 60 ln(θ A) kt + ln A

8 Eample (): (ii)given also that θ A when t, show that k ln ln(θ A) kt +lna At t, θ A ln(a A) k() + ln A ln A k +lna k lna ln A k ln A A k ln Eample (): (iii) Find θ in terms of A when t, epressing our answer in its simplest form. ln(θ A) ln t +lna At t ln(θ A) ln () + ln A θ A e ( ln )()+ln A θ e ( ln )()+ln A + A θ e ( ln )() e ln A + A θ e ( ln ) (A)+A θ e ln( ) (A)+A θ e (ln 9) (A)+A θ 9 (A)+A θ (A)+A θ 7 A APPLICATION ON RATE OF LIQUID LEAKING ON THE BOARD :D 5 6