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1 Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of ths two intgrals to a onstant. Eampl: Solv ( a ) d ( ) d 0 M a ; N ( ) M N M N ; Th givn quation is an at diffrntial quation. Solution is givn b a d d ( ) ( ) 0 a Eampl: Solv ( ) o s d s i n d 0 M Solution is : ( ) Intgrating Fators: N os ; os os d ( ) sin Somtims a diffrntial quations whih is not at ma bom at on multipliation b a suitabl funtion known as an Intgrating Fator. VTU NOTES QUESTION PAPERS
2 For ampl th diffrntial quation d d 0 is not at, but whn multiplid b o r o r b it boms on at quation. Th numbr of intgrating fator is infinit for a givn quation. Ruls For Finding Th Intgrating Fators:: Rul I: whn M N 0 and th quation is homognous thn fator of Md Nd 0 M N is an Intgrating Eampl: Solv: d ( ) d 0 Solution: MN- 4 I.F.-/ 4 d d 0 Multipl Givn quation B I.F. 4 On intgration w hav log Rul II: Whn M - N 0 and th quation has th form f ()df ()d0 thn Intgrating fator is M N Eampl: Solv ( ) d ( ) d 0 M - N I.F./ d d d Multipl givn quation b I.F. w gt on intrgration d log log VTU NOTES QUESTION PAPERS
3 Rul III: Whn N N M is a funtion alon sa f() thn d f ) ( is an Intgrating fator Problm: Solv ( )d d 0 N N M d I.F. Multipling givn quation b I.F. and it boms at Solution of givn quation is givn b ( ) d Rul IV: Whn M M N is a funtion alon sa F() thn d F ) ( is an Intgrating fator Eampl: solv ( ) ( ) d d ( ) ( ) M M N ) (.. F I d F Multipl givn quation b I.F. and it boms 0 4 d d VTU NOTES QUESTION PAPERS
4 Solution is givn b Linar Diffrntial Equation: A diffrntial quation is said to b linar if th dpndnt variabl and its drivativs appar onl in th first dgr. Th first ordr linar diffrntial quation is of th form funtions of onl or onstants. d d P Q whr P and Q ar Solution of Linar diffrntial quation: ( I. F.) Q ( I. F.) whr I.F. Pd d d d Eampl: Solv 4 givn th boundar onditions 0 whn 4. Solution : Rarranging givs d d 4 Intgrating fator pd. Solution givs: ( ) d. Using B.C. Obtain C7/.Hn partiular solution givs ( 7 ). VTU NOTES QUESTION PAPERS 4
5 Eampl: Solv th diffrntial quation d dθ s θ tan θ givn th boundar onditions whn θ 0. d d θ Solution: Rarranging givs ( tan θ ) s θ Intgrating fator ( os θ ) os θ ln pd. θ θ θ θ θ Solution givs: os os ( s ) d. Whn θ 0,, thus. Hn partiular solution is ( θ ). os θ Eampl: Solv th diffrntial quation d d Solution: Intgrating fator pd ln. Solution givs: 4 d VTU NOTES QUESTION PAPERS 5
6 Equations Rduibl to linar form Or (Brnoulli s quation): d d n P Q Dividing b n n d d P n Q ( ) n put v -n w gt ( ) ( ) d d n d i. ( n ) d d ( n ) d ( n ) d Pv Q d ( n ) Pv ( n )Q Whih is linar in v d Eampl: Solv ( ) d Solution: d d ( ) ( ) ( ) Put - v, ( ) d d d VTU NOTES QUESTION PAPERS 6
7 From () i. d d v v whih is linar in v Intgrating fator pd d Solution is v d log Eampl: solv d tan tan d d Solution: tan tan ( ) d Put - v d d d Fron () v tan tan whih is linar in v d Intgrating fator pd tan d log s s VTU NOTES QUESTION PAPERS 7
8 Solution is v s tan s d tan s Orthogonal trajtor Introdution: Th word Orthogonal oms from th Grk right angl and th word trajtor oms from Latin ut-aross. Hn th urv that urv that ut aross th othr at right angl is alld orthogonal trajtor. Ruls to find th quation of orthogonal trajtoris of a famil of artsian urvs: Diff f(,, )0 and liminat. Rsulting quation givs th DE of th famil f(,, )0 i.. d F,. 0 d d Rpla - d d b in F,, 0 d d Th diffrntial quation of orthogonal trajtor is F,,- d 0 d Intgrat F,, d 0 to gt th quation of th rquird orthogonal d trajtor VTU NOTES QUESTION PAPERS 8
9 Ruls to find th quation of orthogonal trajtoris of a famil of polar urvs: Lt th quation of givn famil of urvs b f(r, θ, ) 0 Diff f(r, θ, ) 0 and liminat. Rsulting quation givs th DE of th famil f(r, θ, ) 0 i.. dr F r, θ. 0 d θ Rpla dr b - r d θ dr in F r, θ, 0 d θ dr d θ Th DE of OT is d F r, θ,-r θ 0 dr Intgrat F r, θ, r d θ 0 to gt th quation of th rquird dr orthogonal trajtor Eampl :Find th orthogonal trajtoris of parabolas Solution: Diffrntiat givn quation w gt d C d d d d Rpla w gt is th DE of OT. d d d d d V.S On intgration w gt Eampl: Find th orthogonal trajtoris of th sris of hprbolas k Solution: Diffrntion givn quation w.r.t w gt 0 Rpla b Th diffrntial quation of th OT is d d 0 or - dd d d 0 d d VTU NOTES QUESTION PAPERS 9
10 Whih th rquird OT Eampl: Find th orthogonal trajtoris of irls (-) () solution: Diffrntiation w.r.t, (-) 0 or (-) 0 () From () 0 or ( ) / Substitut this in () w gt () Rpla b Th diffrntial quation of th OT is (4) (5) Put / z Diff w. r. t. Equation (5) boms linar in z I.F./ solution is 0 or 0-0 or d d d d or b 0dividn d d d dz d d d dz z 0 or or d z VTU NOTES QUESTION PAPERS 0
11 Or ( ) b th quation of OT Eampl: Th orthogonal trajtoris of ardiod ra(os(θ)) whr a is a paramtr. Solution: Givn quation r a(os(θ)) Diff. w.r.t. θ dr asinθ a dr dθ sinθ dθ r sin θ os θ r dr sin θ dθ dθ os θ θ os θ ( os θ ) dr r sin θ r tan Whih is th DE of famil ra(os(θ)) dr θ dr θ Rpla ot dθ Intgrat this w gt dθ r ot sin θ log r log on simplifiation r (- os(θ)) VTU NOTES QUESTION PAPERS
12 VTU NOTES QUESTION PAPERS
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