Differential Equations

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Wesley Howard
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1 UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs with siz of th radius, th bnding of a bam changs with th wight of th load placd on it, th path of a projctil changs with th locit and angl at which it is fird. In th languag of mathmatics, changing ntitis ar calld ariabls and th rat of chang of on ariabl, with rspct to anothr a driati. Equations which prss a rlationship among ths ariabls and thir driatis ar calld diffrntial quations. But hr w ar intrstd to know that how th ariabls thmsls ar rlatd. For ampl, from crtain facts about th ariabl position of a particl and its rat of chang with rspct to tim, w wish to dtrmin how th position of th particl is rlatd to th tim so that w can know whr th particl was, is, or will b at an tim t., Diffrntial quations thus originat whnr a unirsal law is prssd b mans of ariabls and thir driatis.. DIFFERENTIAL EQUATION An quation which contains th diffrntial cofficint is calld diffrntial quation, whr is dpndnt and is indpndnt ariabl. For ampl, (i) d d d 4 (ii) Ordr and Dgr of Diffrntial Equation Th ordr of th highst driati of an diffrntial quation is calld ordr of th diffrntial quation. Th dgr of th highst ordr driati aftr rmoing th radical sign and fraction is calld dgr of th diffrntial quation.

2 A TEXTBOOK OF ENGINEERING MATHEMATICS II In ampl (i), th ordr is on and dgr is also on, and in ampl (ii) th ordr is two but dgr is on. Thr ar som dfinitions of constants which us in th solution of diffrnial quation. Constant: A smbol, which rtains th sam alu during a st of mathmatical oprations, is calld a constant for ths oprations. Absolut Constant: A constant, which has th sam alu in r mathmatical opration, is calld an absolut constant. For ampl, 5, 6, p,,... ar absolut constants. Arbitrar Constant: A constant, which has a fid alu throughout a particular rfrnc and dos not rmain fid in gnral, is calld an arbitrar constant. For ampl, m c is th quation of a straight lin in which m and c ha som fid alus for a particular straight lin but ha diffrnt fid alus for anothr straight lin. Hr m and c ar calld arbitrar constants... Solution of Diffrntial Equation An quation inol dpndnt ariabl () as wll as indpndnt ariabl () and fr from driati, which satisfis diffrntial quation, is calld th solution of th diffrntial quation.g., C C is th solution of th diffrntial quation d.. Gnral Solution of Diffrntial Equation 0....(A) Th solution in which th numbr of arbitrar constants ar qual to th ordr of diffrntial quation is calld gnral solution of diffrntial quation...4 Particular Solution of Diffrntial Equation Th solution for an particular alu of constants is calld particular solution of diffrntial quation. If C, C thn is th particular solution of diffrntial quation (A). Linar Indpndnc and Dpndnc Two functions () and () ar said to b linarl indpndnt if C C 0...(i) implis C C 0 i.., or cannot b prssd as proportional to othr. If C C 0, 0 thn () and () ar linarl dpndnt.. ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER A diffrntial quation of first ordr classifid in th following diffrntial quations: (i) Gnral form (Variabl sparation mthod) (ii) Homognous diffrntial quations (iii) Linar diffrntial quation of th first ordr (i) Eact diffrntial quation... Variabl Sparation Mthod In this mthod w sparat th ariabls i.., on lft hand sid and on right hand sid thn intgrat on both sids.

3 DIFFERENTIAL EQUATIONS Eampl : Sol th diffrntial quation ( 7) ( 8) d 0. (U.P.T.U. 005) Solution: Th gin diffrntial quation can b writtn as 8 d 7 Appling componndo and diidndo rul, w gt d 5 d d 5 5 5( ( d 5 Multipling on both sids b, w gt d Intgrating on both sids, w gt d 5 log( ) 5log( ) logc 5 C ( ). Ans. Eampl : Sol th diffrntial quation ) ) ( )cos sin d 0. Solution: Th gin diffrntial quation can b writtn as ( ) cos sin d cos d ( ) sin d cos ( ) logc sin log ( ) log sin logc log ( ) logsin logc log ( ) sin logc ( )sin C. Ans... Homognous Diffrntial Equations d f (, ) A diffrntial quation of th form, whr f (, ) and g (, ) ar homognous functions g(, ) in and of th sam dgr is calld homognous diffrntial quation. Such tp diffrntial quations ma b sold b putting

4 4 A TEXTBOOK OF ENGINEERING MATHEMATICS II d, so that And th quation rducs to th ariabl sparabl form. d Eampl : Sol. Solution: Th gin quation can b writtn as d d It is a homognous, hnc putting, w gt ( ) logc ( ) ( ) ( ) log logc log C logc logc log log log C log logc C C C. Ans.

5 DIFFERENTIAL EQUATIONS 5 d Eampl 4: Sol tan. d Solution: d Putting, w gt, tan rplac b, tan tan tan tan C cot C log sin C So, log sin C. Ans.... Equations Rducibl to Homognous Form Th diffrntial quations undr this catgor ar of th form d a b c...(i) a b c whil rducing this quation in homognous form thr aris two cass. a b Cas I: Whn, a b d dy w substitut X h, Y k hnc, (i) bcoms dx dy a ( X h) b ( Y k) c ( a X by ) ( ah bk c)...(ii) dx a( X h) b ( Y k) c ( ax by) ( ah bk c ) whr h and k ar som constants choosn such that a h bk c 0, a h bk c 0 Thn quation (ii) rducs in th form dy a X by dx a X by which is a homognous. Th alus of h, k obtaind from abo two quations. a b Cas II: If, in this cas w cannot calculat th alu of th constants h and k. So, a b w ha b a k a a k b b k and a b

6 6 A TEXTBOOK OF ENGINEERING MATHEMATICS II Hnc th quation (i) bcoms d ( a b ) c k( a b ) c Hr w substitut a b z and it can b sol b ariabl sparation mthod. d Eampl 5: Sol. 6 4 Solution: a b a b and a 6 b 4 a b Th gin quation ma b writtn as d ( )...() or d dz d dz Lt z Using this alu in quation (), w gt dz z dz 4z z z z dz 4z dz z 4z z z C (4z ) dz z C log(4z ) C 8 z log4 log z Now, rplac z b, w gt C or log ( ) C log 4 C. Ans. Eampl 6: Sol Solution: d. 5 d 5...()

7 DIFFERENTIAL EQUATIONS 7 a b a,, b so lt X h and Y k Thn th quation (), rducs in th form dy dx ( Y k) ( X h) ( Y X) ( k h ) ( Y k) ( X h) 5 ( Y X) ( k h 5) Lt us choos h and k such that k h 0, k h 5 0 which gi h, k, thn (), bcoms Putting Y X, w gt dy dx Y Y X X which is homognous. X ( ) X X dx X ( ) dx X dx ( )...() dx, on intgrating X dx log C X log( tan Y X tan ) tan log X logc log Y X log log X logc Y X logc 0 0 tan log ( ) ( ) logc 0 As X, Y tan log C ( ) ( ) 0. Ans... Linar Diffrntial Equation of th First Ordr A diffrntial quation of th form d P Q...(i)

8 8 A TEXTBOOK OF ENGINEERING MATHEMATICS II is said to b a linar diffrntial quation of th first ordr, whr P and Q ar functions of onl or constants. It was introducd b Libnitz and is thrfor bttr known as Libnitz s linar diffrntial quation. Mthod of Solution: To sol it, w multipl (i) on both sids b a factor P, w gt Intgrating both sids, w gt d P d P P P P P Q C Q Q P P which is th rquird solution whr th factor P is calld intgrating factor. Rmark: If th quation is of th form P Q, thn solution of this quation, will b as d follows: Pd Pd Q d C whr P and Q ar th functions of onl. Eampl 7: Sol ( log ) ( log ) d 0. (U.P.T.U. 004) Solution: Th gin quation ma b writtn as. which is linar in. d log Hr P, Q log I.F. Pd d log log Th solution is log log d C log (log) C. Ans. d Eampl 8: Sol ( ) ( ). Solution: Th gin quation ma b writtn as (A.M.I.E.T.E. 00)

9 DIFFERENTIAL EQUATIONS 9 d ( ), which is th linar in, Hr, P and Q ( ) Hnc, th solution is I.F. P ( ) C C. Ans. log( ) log( )... Equations Rducibl to Linar Form (Brnoulli s Equation) d n Th diffrntial quation P Q...(i) whr P and Q ar functions of onl is known as Brnoulli s quation. Mthod rducd to linar form: Diiding quation (i) b n on both sids, w gt d P Q...(ii) n n Putting z, so that n ( n) d dz d dz n n ( n) Substituting ths in (ii), w gt dz dz Pz Q ( n) Pz ( n) Q ( n) which is linar in z and can b sold asil. Not: Th quations of th form d P. f ( ) Qf () ma b also sol b rducibl to linar form, whr P and Q ar functions of onl or constant. d Eampl 9: Sol. Solution: Th gin quation ma b writtn in th form d d

10 0 A TEXTBOOK OF ENGINEERING MATHEMATICS II...() d Diiding on both sids b, w gt Lt d z Substituting this alu in quation (), w gt dz z d d dz d d dz d...() or dz z d, which is linar in z....() Hnc P, Q I.F. d log log Th solution is z d C z z log C 0 log C log C 0 As z log C 0; whr C C. Eampl 0: Sol d ( ) 0. (U.P.T.U. 006) Solution: Th gin quation ma b writtn in th form d Diiding on both sids b, w gt d....() Lt z, dz so that. d

11 DIFFERENTIAL EQUATIONS Substituting in quation (), w gt which is linar in z. Hnc, P, dz z Q dz z Th solution is I.F. I.F. z log log C z C z C..4 Eact Diffrntial Equation C As z C. Ans. A diffrntial quation of th form M Nd 0, whr M and N ar functions of, is said to b act diffrntial quation if it satisfis th condition Mthod of Solution: (i) First of all show that (ii) Th solution is Eampl : Sol const. M N(Th trms which do not contain ) d C ( sin cos) ( sin cos) d 0. Solution: Comparing with M Nd 0, w gt M sin cos N sin cos

12 A TEXTBOOK OF ENGINEERING MATHEMATICS II \ sin cos cos sin and This shows that Th gnral solution is sin cos cos sin, so th gin quation is act M N(trms do not contain ) d C ( sin cos) 0. d C cos cos C sin sin cos C cos sin cos C sin C. Ans. / / Eampl : Sol ( ) d 0. Solution: Comparing th gin quation with M Nd 0 M / / ( ), N and \. / / / / / / / Sinc

13 DIFFERENTIAL EQUATIONS Th solution is ( ) d 0 / 0 / C. Ans...4. Rduction of Non-act Diffrntial Equations into Eact Diffrntial Equations If th diffrntial quation M Nd 0 is not act. It ma bcom ist on multipling b a suitabl function known as intgrating factor. Thr ar som ruls for finding out intgrating factor. Rul For homognous quation If M N d 0 is a homognous quation in and thn I.F. M N Eampl : Sol ( ) ( ) d 0 Solution: Th gin quation is homognous. Hr, M and N \ and Sinc, th quation is not act. \ I.F. M N ( ) Multipling th gin D.E. b I.F., w gt ( ) ( ) d ( ) ( ) 0 d 0...() Hr, M ( ) and N ( ) \ \ Equation () is act ( ) and ( )

14 4 A TEXTBOOK OF ENGINEERING MATHEMATICS II Hnc, th solution is Hnc, d C log log( ) log C log log( ) C log ( ) C ( ) C C. Ans. Rul For th form f () f () d 0 If th quation M Nd 0 is of th form f ) f ( ) d 0 ( thn I.F. M N Eampl 4: Sol ( sin cos) ( sin cos) d 0 Solution: Hr, M sin cos and N sin cos \ sin cos cos sin or sin cos cos and or This shows that \ I.F. sin cos cos sin sin cos cos Multipling th gin quation b I.F., w gt sin cos sin cos cos tan d tan 0...()

15 DIFFERENTIAL EQUATIONS 5 Hr, M tan and N tan \ tan sc and N tan sc, th quation () is act. Hnc, th gnral solution is d tan C logsc log log C sc log sc C log C Rul sc log log C whr, C log C sc C. Ans. In th quation M Nd 0 If ƒ() [function of onl] N thn I.F. f ( ). Eampl 5: Sol ( ) d 0 Solution: Hr M and N \ and Now N ( ) 4 4 \ I.F. f( ) 4 / 4log 4 I.F. 4 log

Southern Taiwan University

Southern Taiwan University Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution: