MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous and linar diffrntial quations of th tp p() q() Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
C Ordr and Dgr of a Diffrntial Equation : Ordr : Ordr th highst diffrntial apparing in a diffrntial quation. MD C O N C E P T S Dgr : It dtrmind b th dgr of th highst ordr drivativ prsnt in it aftr th diffrntial quation clard of radicals and fractions so far as th drivativs ar concrnd. m d f (, ) m n m d f (, ) n nk...fk (, ) m Th abov diffrntial quation has th ordr m and dgr n. Practic Problms :. Th ordr of th diffrntial quation whos gnral solution givn b = (c + c ) + c. Th dgr of th diffrntial quation, of which = a( + a) a solution,. Th ordr of th diffrntial quation whos gnral solution givn b : c = c cos( + c ) (c + c ) a 5 c6 sin( c 7 ) 5 [Answrs : () d () b () c] 0 c C Formation of Diffrntial Equation : Th diffrntial quation corrsponding to a famil of curv can b obtaind b using th following stps : Idntif th numbr of ssntial arbitrar constant in quation of curv. If arbitrar constants appar in addition, subtraction, multiplication or divion, thn w can club thm to rduc into on nw arbitrar constant. Diffrntiat th quation of curv till th rquird ordr. Eliminat th arbitrar constant from th quation of curv and additional quation obtaind in stp abov.. Th diffrntial quation of th famil of curvs = (A cos + B sin ), whr A and B ar arbitrar constants, d d 0 0 d 0 [Answrs : () c] C Solution of a Diffrntibl Equation : Th solution or th intgral of a diffrntial quation, thrfor, a rlation btwn dpndnt and indpndnt variabls (fr from drivativs) such that it satfis th givn diffrntial quation. Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
Practic Problms :. Th solution of th diffrntial quation, + = 0 MD constant constant constant non. Th solution of th diffrntial quation; ( + / ) = ( / ) / = ln( / + ) = ln( / + ) + = ln( / + ) = ln( / + ) [Answrs : () a () b] C Solution of lmntar tps of first ordr and first dgr diffrntial quations : Equations Rducibl to th Variabls Sparabl form : Its gnral form = f(a + b + c) a, b 0. To solv th, put a + b + c = t. Homognous Diffrntial Equation : C5 A diffrntial quation of th form f(, ) whr f and g ar homognous function of and, g(, ) and of th sam dgr, calld homognous diffrntial quation and can b solvd asil b putting = v. Equations Rduciabl to th Homognous form Equations of th form a b c A B C can b mad homognous (in nw variabls X and Y) b substituting = X + h and = Y + k, whr h and k ar constants. Now, h and k ar chosn such that ah + bk + c = 0, and Ah + Bk + C = 0; th diffrntial quation can now b solvd b putting Y = vx. C6 Linar diffrntial quations of first ordr : Th diffrntial quation P Q, linar in, whr P and Q ar functions of. Intgrating Factor (I.F.) : It an prssion which whn multiplid to a diffrntial quation convrts it into an act form. I.F. for linar diffrntial quation = P (constant of intgration will not b considrd) d Aftr multipling abov quation b I.F. both sid, it bcoms (. P ). P Q. P + c. = Q P Som tims diffrntial quation bcoms linar if takn as th dpndnt variabl and as indpndnt variabl. Th diffrntial quation has thn th following form : Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
MD P Q Practic Problms :. whr P and Q ar functions of. pth I.F. now. Solution of th diffrntial quation ( ) P = (c + ) = (c ) = (c ) = (c + ). Th quation of th curv whos tangnt at an point (, ) maks an angl tan ( + ) with -a and which passs through (, ) 6 + 9 + = 6 ( ) ( ) 6 9 + = 6 6 + 9 = 6 ( ) Non of ths [Answrs : () d () a] C7 Tim saving tips :. Th quation of th form n P Q whr P and Q ar functions of onl and n constant n n {n 0, } can b rducd to linar form b. P. Q and put n = v.. Som important intgrs factors (I.F.) for th quick solutions (i) d tan (ii) dlog (iii) d( m n ) = m. n.(m + n) (iv) d log Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
MD 5 INITIAL STEP EXERCISE. Th dgr of th diffrntial quation 5/ 5 5 d : 5 0. Th diffrntial quation rprsnting th famil of curvs = c ( + c) whr c a positiv paramtr of ordr ordr dgr dgr. A solution of th diffrntial quation 0 = = = =. Th solution of th diffrntial quation = ( + ) + ( ) = ( + ) = ( + ) + = ( ) + = log ( ) 5. For solving, suitabl substitution = a = + v = + + = v 6. Th solution of th diffrntial qual ( + ) + = 0 ( + c) + = 0 ( c) + = ( c) + + = 0 7. Th solution of th diffrntial quation [ sin ] (cos sin c ) = (cos sin c ) = (cos sin / ) = non of th abov 8. If () a diffrntiabl function, thn th solution of + ( () () ()) = 0 = (() ) + c () () = ( () + c () = () () + c ( ()) = ( ()) () 9. Th solution of th diffrntial quation cos tan c tan cos c (c c ) tan 0. Solution of c = c c c. Th ordr of th diffrntial quation whos gnral solution givn b c = (c + c ) cos ( + c ) c 5 5. Th solution of th quation ( ) ( ) FINAL STEP EXERCISE ( ) c c c ( ) c Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
. Th curv for which th normal at an point (, ) and th lin joining origin to that point form an ascls triangl with th -a as bas an llips a rctangular hprbola a circl non of th abov. Th quation of th curv in which th portion of th tangnt includd btwn th co-ordinats as bctd at th point of contact a parabola an llips a circl a hprbola 5. Th quation of th curv which such that th portion of th a of cut off btwn th origin and tangnt at an point proportional to th ordinat of that point = (a b log ) log = b + a = (a b log ) non of th abov 6. Th solution of / d ( / + ) = 0 / + = c / + = c / + = c / + = c 7. Th curvs for which th lngth of th normal qual to th lngth of th radius vctor ar onl circls onl rctangular hprbola ithr circl or rctangular hprbola non of th abov 8. Th famil of curvs rprsntd b and th famil rprsntd b MD 6 touch ach othr ar orthogonal ar on and th sam non of th abov 9. Th orthogonal trajctivs of th famil of curvs a n = n ar givn b : n + n = constant n + = constant n + n = constant n n = constant 0. Th quation of th curv passing through th point (, ) and satfing th diffrntial quation ( ) + ( + ) d = 0 log = log = + log log. If m and n ar th ordr and dgr of th diffrntial quation d d 5 d m =, n = m =, n = m =, n = 5 m =, n = d, thn 0. ANSWERS (INITIAL STEP EXERCISE) ANSWERS (FINAL STEP EXERCISE). b. a. b. d 5. d 6. d 7. b 8. a 9. c 0. b. c. b. b. d 5. c 6. c 7. c 8. b 9. b 0. a. b Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
MD 7 AIEEE ANALYSIS [00] d. Th solution of th quation c c d d. Th diffrntial quation of all non-vrtical lins in a plan d 0 0 d 0 0 c d. Th ordr and dgr of th diffrntial quation / d ar, (, ) (, ) (, ). Th dgr and ordr of th diffrntial quation of th famil of all parabolas whos a -a, ar rspctivl,,,, 5. Th solution of th diffrntial quation ( ) ( tan ) 0 AIEEE ANALYSIS [00] tan tan tan tan ( ) k tan tan k k tan k AIEEE ANALYSIS [00/005] 6. Th diffrntial quation for th famil of curvs + a = 0, whr a an arbritrar constant ( ( ( ) ) ) 7. Th solution of th diffrntial quation + ( + ) = 0 log C log C [00] ( ) [00] C log = C Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
8. If (log log ), thn th solution of th quation log c log c MD 8 9. Th diffrntial quation rprsnting th famil of curvs = c c, whr c > 0, a paramtr, of ordr and dgr as follows ordr, dgr ordr, dgr ordr, dgr ordr, dgr [005] [00] log c log c AIEEE ANALYSIS [006] 0. Th diffrntial quation whos solution A + B =, whr A and B ar arbitrar constants of scond ordr and first dgr scond ordr and scond dgr first ordr and scond dgr first ordr and first dgr AIEEE ANALYSIS [007]. Th diffrntial quation of all circls passing through th origin and having thir cntrs on th -a. Th normal to a curv at P(, ) mts th -a at G. If th dtanc of G from th origin twic th abscsa of P, thn th curv a circl hprbola llips parabola ANSWERS AIEEE ANALYSIS. d. d. a. a 5. a 6. a 7. b 8. d 9. d 0. a. a. b Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857
MD 9 TEST YOURSELF. Th solution of = ( + ), () = tan (/) + log ( + ) = log tan (/) + log ( + ) = / + log tan (/) + log ( + ) = log. A curv passing through origin, all th normals to which pass through ( 0, 0 ) 0 = + + = ( 0 + 0 ) + = 0 + 0. Th gnral solution of 5 = givn b = C 5 + C + C + C + C 5 = C 5 + C + C + C + C 5 = C + C + C + C + C 5. Th curv such that th product of th dtancs from an tangnt to two givn points constant rprsnts circls straight lins llips and hprbolas 5. Th gnral solution of 8. Solution of = C + C = C + C = C + C = C + C / 9. Gnral solutions of / C = / log C = / log C = / log C = / 0. Th solution of th diffrntial quation d whn (0) = and (0) = 0 6 9 9 ( ) 8 6 9 9 ( ( / ) / ) = ( / ) = ( / ) = ( / ) = C ( / ) 6. Th gnral solution of satfing () = = (/) (/) = (/) (/) = 7. Th diffrntial quation of a curv such that th initial ordinat of an tangnt at th point of contact qual to th corrsponding subnormal a linar quation not a homognous quation an quation with sparabl variabls. b. b. a. c 5. b ANSWERS 6. c 7. a 8. c 9. b 0. b Einstin Classs, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outr Ring Road Nw Dlhi 0 08, Ph. : 96905, 857