DIFFERENTIAL EQUATIONS

Transcription

1 Matheatic Reviion Guide Introduction to Differential Equation Page of Author: Mark Kudlowki MK HOME TUITION Matheatic Reviion Guide Level: A-Level Year DIFFERENTIAL EQUATIONS Verion : Date: 3-4-3

2 Matheatic Reviion Guide Introduction to Differential Equation Page of Author: Mark Kudlowki DIFFERENTIAL EQUATIONS A differential equation in ter of the variable and i one that contain a differential coefficient, nael a ter like (or If onl the firt derivative i preent, a in the eaple above, we have a firt-order differential equation Second-order differential equation (beond the cope of A-level) would include the econd derivative, ie d (or Eaple of firt-order differential equation are =, = and = If =, then integrating with repect to give = + c Thi i the general olution of the differential equation, and it can be illutrated b a fail of curve Each curve ha the equation = + c, with the cae where c = hown in bold A particular olution i one pecific curve of the fail of general olution, and it can be found fro additional inforation Eaple (): Find the particular olution of the differential equation when = = 4 + 3, given that = 8 Integrating uing tandard ethod give a general olution of = c Subtituting = and = 8 give c = 8 and hence c = -6 The particular olution i therefore = + 3 6

3 Matheatic Reviion Guide Introduction to Differential Equation Page 3 of Author: Mark Kudlowki Eaple (): Find the particular olution of the differential equation when = = co, given that = 3 Integration give a general olution of = in + c Subtituting = and = 3 give + c = 3 and hence c = 5 The particular olution i therefore = in + 5 The above eaple were both olved uing tandard technique learnt earlier in the coure, but to tackle an equation like =, we need a new ethod Separating the variable If a differential equation can be written in the for f() = g(), then it can be olved b eparating the variable B treating and a if the were algebraic quantitie and integrating both ide, thi epreion can be rewritten a f ) g( ) ( Eaple (3): Solve = Thi epreion ut firt be anipulated to eparable for = = Note that the tage = i atheaticall nonenical and i onl hown a a tep in the working we ut prefi integral ign to give the epreion eaning! Integrating, we have ½ = ½ + c Thi can be rewritten a = + c (Since c i an arbitrar contant, we till ue c and not c after the doubling) The contant could have been attached to either ide of the epreion Eaple (4): Solve =, given that = when = 3 Manipulating to eparable for we have = Integrating, we have the general olution ½ = + c Thi can be rewritten a = 4 + c Therefore, = 4 + c Uing the reult = 4 + c and ubtituting = 3, =, we have 4 = + c c = -8 The particular olution of the differential equation i = 4 8

4 Matheatic Reviion Guide Introduction to Differential Equation Page 4 of Author: Mark Kudlowki In the lat eaple, we had rewritten general olution uch a ½ = + c in the for = 4 + c There wa no need to write = 4 + c, a both c and c were ipl nuber Siplifing the arbitrar contant in thi wa i allowable onl when baic arithetic i applied to both ide of the differential equation Care ut be taken with the arbitrar contant if an atheatical operation other than baic arithetic are ued, uch a raiing to power or taking eponent Eaple (5): i) Find the general olution of = ii) Find the particular olution of the ae equation, given initial condition of = and = 5e Separating the variable, = Thi i a pecial cae of the for f() = g() where g() = Integrating we have or ln = + c Taking eponential of both ide we have = e (+c) Thi lat epreion can be rewritten a = Ae where A = e c Notice how the arbitrar contant tarted off a a quantit to be added, but when we took eponential, the contant wa tranfored into a ultiplier b the log law The contant ut be incorporated iediatel after perforing the original integral It can be added to either ide practice will tell ou where it i ore uitable to place it! Thu we could have aid or ln + c = Taking eponential of both ide we have e c = e, and then A = e where A = e c The forer arrangeent i ore logical and i the one uuall quoted in tetbook ii) Subtituting = and = 5e into the general olution give A = 5, and hence the particular olution for the differential equation i = 5e NB The ethod below i totall incorrect! It i ver bad practice to diregard the arbitrar contant throughout the working and jut tag it on at the end, a hown b the wrong ethod below Begin with, then ln =, then take eponential to get = e, and finall, a an afterthought, tag the contant on at the end to obtain = e + c Subtituting = and = 5e into the incorrect general olution give C = 4e, and the incorrect particular olution for the differential equation of = e + 4e

5 Matheatic Reviion Guide Introduction to Differential Equation Page 5 of Author: Mark Kudlowki Eaple (6): i) Solve the differential equation co t, to find in ter of t ii) Given the initial condition of t = and = 4, how that the olution can be written in the for in at b where a and b are poitive contant to be deterined i) Manipulating to eparable for we have co t t co t t in c (halving both ide) in c (quaring both ide) Note how we did not need to a t in c both ide, we had to put the contant inide the bracket and not a t in c when we halved both ide, but when we quared t in c ii) Given that = 4 when t =, we ubtitute into the general olution in 4 Hence the particular olution of the differential equation i in 5t Eaple (7): Solve the differential equation 4 4 (taking eponential) e c c to give c = 4, given initial condition of =, = 5 ln c Ae where A = e c Subtituting the initial condition of =, = 5 5 Ae 5 A The particular olution i therefore 5e Eaple (8): Solve the differential equation ( 3)(4 3), given initial condition of = -, = 3(e - +) ( 3)(4 3) ln( 3) 3 c (taking eponential) 3 e 3c Initial condition: = -, = 3(e - c +) e 3 3 e 3e e c (taking natural log) (ln 3) 3 3( e ) c c ln 3 3 3e 3

6 Matheatic Reviion Guide Introduction to Differential Equation Page 6 of Author: Mark Kudlowki Eaple (9): Solve the differential equation e in, given initial condition of = /, = e in e in e in e co c A = /, =, c = e Hence e co e and ln(co e) Eaple (): Solve, given initial condition of =, = Before continuing with the proce, we need to iplif the bracketed ter into an integrable for, uing partial fraction A B ( ) ( ) Partial fraction decopoition: A(-) + B(+) = A + B = (equating ter) ; B - A = (equating contant) A = - ; B = We now continue with Separating variable, ln ln ln c Ae (eponentiating both ide) (A = e c ) Subtituting =, = give A and hence A = - The particular olution to the differential equation i therefore e or e

7 Matheatic Reviion Guide Introduction to Differential Equation Page 7 of Author: Mark Kudlowki Eponential Growth and Deca If we were to repeat eaple (5) uing = k and = -k where k i a poitive contant, we have eponential growth and deca function Becaue eponential growth and deca take place over an interval of tie, it i ore coon to ue t for a the variable with repect to which we are differentiating

8 Matheatic Reviion Guide Introduction to Differential Equation Page 8 of Author: Mark Kudlowki One everda eaple of eponential growth i that of a population where p p e kt dp where t i the elaped tie and p i the population at the tart of the tie interval Thi forula can alo be applied to copound interet proble where the interet i copounded continuoul (to give another eaple) kp Eaple of eponential deca include: Diintegration of radioactive aterial where kt k e where t i the d elaped tie and i the a of the aterial at the tart of the tie interval Newton Law of Cooling, where d k ( ) ( ) e where t i the elaped tie, i teperature of the object, i the teperature of the urrounding, and i the teperature of the object at the tart of the tie interval Eaple (): The population of the United Kingdo wa etiated at 57 illion in Auing an annual rate of growth of 5%, what will be the population etiate for? Thi proble can be epreed a a differential equation, ie kt dp kp p p e kt where p = 57, t = and k = 5 We want to find p uch that p p e kt or p 57e 5 p = 763 illion (Note that thi i not the ae a 57 (5) or 745 illion, ince here the growth i continuou, and not jut at the end of each ear!) Eaple (): The radioactivit of a aple of ulfur-35 wa taken in ter of a Geiger counter reading, and the adjuted count wa recorded a 85 hit per inute The ae aple wa then locked awa and another reading taken 8 da later, and then the adjuted count wa recorded a 485 hit per inute Auing that the nuber of hit i proportional to the a of the ulfur-35 reaining, what i the dail deca contant k and hence the half-life of ulfur-35? (The half-life of a radioactive ubtance i the tie taken for the a, and hence the activit, to decline to one-half of it original value) d kt k e aterial at the tart of the eperient where t i the elaped tie (in da here) and i the a of the Here k i the deca contant which i unknown, = 85, = 485, and t = 8 (The nuber of hit in each cae repreent the a) The deca contant k i found b kt e e kt

9 Matheatic Reviion Guide Introduction to Differential Equation Page 9 of Author: Mark Kudlowki Taking log of both ide k ln t ln485 ln85 k 8 -k = -789 To find the half-life, we ue We ut olve e kt and find the value t which ake kt e kt ln (taking log) ln kt ln t k the half-life of ulfur-35 i ln 789 da or 878 da

10 Matheatic Reviion Guide Introduction to Differential Equation Page of Author: Mark Kudlowki Eaple (3): Tea wa poured into a vacuu flak and the teperature wa recorded a 95 C iediatel before the flak wa toppered and ealed After one hour the teperature of the tea wa eaured a 76 C Given an eternal teperature of C, for how long would the flak have been epected to keep the tea hot? ( Hot ean a teperature of 5 C or above) We ue Newton Law of Cooling here: d k kt ( ) ( ) e where t i the elaped tie (in hour here), i teperature of the tea after one hour, i the teperature of the urrounding, and i the tarting teperature of the tea We want to ake k the ubject of ) kt ( e : e kt kt ln (take log) kt k ln (logarith of a reciprocal i tie the logarith of the nuber) ln( ) ln( ) (log law) t Subtituting t = (elaped tie in hour), 76 (teperature of tea after hour) = (urrounding teperature), and = 95 (teperature at tart), we have ln( 83) ln(64) k = 6 Having found k, we can therefore find the value of t correponding to a tea teperature of 5 C To do thi we rearrange the lat forula to ake t the ubject and ubtituting 5 t ln( ) ln( ) t or 467 hour k ln( 83) ln(38) 6 the tea can be epected to reain hot (above 5 C) for about 4 hour and 4 inute