Unit-I Linear Differential Equations
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1 Enginring Mathmatics-III Unit-I Linar iffrntial Equations Linar iffrntial Equations An quation in th form a n n- + a +...a + a n- + a 0 = X n n n whr a 0, a, a,..a n ar constants an X is a function of is call a highr orr iffrntial quation. If = thn th abov quation can b rss as c a + a + a +...a + a + a = X b g b g n n n- n- n- a- 0 i.. f = X whr f is a ol nomial in '' of gr n. with constant cofficint. Th solution of such an quation is givn b = c + whr c call th Comlimntar function an is call th Particular intgral. h A. Mtho of Fining C.F. : Consir th auiliar quation f() = 0. Now f() can in gnral b factoriz into n factors an th abov quation can b rss as (-m ) (-m ) (-m ) (-m n ) = 0. Comlimntr function ns uon natur of roots of auillar quation. i) If m, m, m,.m n ar all Ral an istinct m m m m m n c n thn = c + c + c + c +...c ii) If a root (sa) m is rat twic thn that art of th solution bcoms c = (c + c ) m If m is rat thric, thn that of th solution bcoms c = (c + c + c ) m an so on. iii) If th roots ar imaginar (sa) i, thn th solution bcoms c = a (c cos + c sin). iv) If i, ar rat twic, thn th solution bcoms c = [(c + c ) cos +( c +c ) sin] B. For fining P.I. : In gnral, = f b g X ning on X w hav iffrnt mthos for fining Prof. M G Shrigan Pag
2 Enginring Mathmatics-III Linar iffrntial Equations T I] If X = a thn = f b g f a a b g b g a f Rlac '' b 'a' in f rovi f a 0 If f(a) = 0 thn rss f() as f() = () (-a) r whr () os not contain th factor (-a) [i.. (a) 0] Thn = a = - a r a r! b g b g a f T II] If X = sin (a + b) or cos (a + b) thn sinca + b h = sin ca + b f f -a j j [Rlac b a in f( ) rovi f(-a ) 0 If f -a 0 an th factor + a is rat 'r' tims in f thn j j j a j r sin a + b = - r c h b g b r! a g r r F HG h r a sin a + b + r In articular if r =, thn sin a + b = - sin a + b = cos a + b c h c h c h + a a cos a + b = + a cos a + b = sin a + b c h c h c h a T III] If X = m whr m is a ositiv intgr thn = f = = b g b g gbg bg m m m n whr () is a function of or constant Ean [ g ()] -n b Binomial Eansion an solv b iffrntiation. Powrs of highr thn m n not b consir. I K J Not b b b b g g g g = = = = Prof. M G Shrigan Pag
3 Enginring Mathmatics-III n b g b g b g b g In gnral + = - n + n n - + n n - n n b g b g b g b g - = + n + n n - + n n - n Linar iffrntial Equations T IV] If X = a V whr V is som function of thn = f a V b g a f + a b g a V [Rmov a outsi an thn rlac b + a in f() an thn solv it b on of th rvious t]. T V] If X = V whr V is som function of Thn [V ] f () ' f () f () f () T VI] Gnral Mtho Invrs orator V whr V is som function of m m -m z m V =. V V Not z m -m V = m. V fa f = z fa f, f = f, f f = c h zz a f a f a f zzz. (c-0) Gnral Mtho. (c-08,09,0) Ans: C C log. (c-07)(ma-0) Ans.: C C log log Prof. M G Shrigan Pag
4 Enginring Mathmatics-III Linar iffrntial Equations. (Ma-07) 5. Ans.:C C log ( ) cos (c-07,0,) 6. (Ma-0) 7. sin cos (c-0) (Ma-) 8. (c-0,) (Ma-05) 9. sin (c-09) (c-08). cosc (c-0). 9 sc (Ma-08) Short-cut Mthos Rul No: 5. (Ma-08) Ans.: C C C. (c-07) Ans log (c-0) : C C 6. 6 (Ma-07) (c-) 7. (c-05) 5 log log5 9 Ans.: C C C C.: C C C C log Ans log Prof. M G Shrigan Pag
5 Enginring Mathmatics-III Linar iffrntial Equations sinh (Ma-) Ans: C C cos sin 9. sinh (c-0) 0. cosh (c-0). (c-07) Ans : C Ans: C C C C 0 C Cosh Ans.: C C C sinh 8 sinh 6.. cosh sinh (Ma-05) Ans: c c cos c sin cosh5 80 cosh 8. ( ) (c-07). ( ) ( ) (c-) E : Solv Solution : Th 8 (MAY 007) (c 0) A.E. is 0, 8 0 c c c log 0 log 0 7 () 8 log 0 (log ) log log 8 Th gnral Solution is givn b, (log ) 8 c c c 7 (0) 8 Prof. M G Shrigan Pag 5 0 log log 8
6 Enginring Mathmatics-III Linar iffrntial Equations Rul No: cos cos (Ma-0) 5. 5 Ans:C cos C sin C cos C sin Ans 6. cos (c-09) :C C sin C cos 5 7. sin (Ma-08) Ans: C 6 C C cos C C sin cos sin 8. Sin (c-07) 5 9. Cos (Ma-07) 0. sin (c-0) Ans: C C 5 C cos C sin 9 sin 90 sin 5 sin cos.: C cos C sin cos sin Ans Ans.: C 6 C C cos C C sin cos sin 5 cos 8. sin 5 (c-08). sin (Ma-09) (c-0). cos.cos (c-0). sin cos (c-09) 5 Ans :C C cos C sin sin Ans.: C C cos C sin sin 8 Ans:C cos C sin 6 cos Ans:C cos C sin sin 5 8 sin log log cos 5. sin sin (c-) (Ma-07) 6. sin 5cos (c-09) 7. 6 cos (Ma-09) Ans.: C cos C sin 6 cos Ans : C C C cos C sin 6 sin sin Prof. M G Shrigan Pag 6
7 Enginring Mathmatics-III 8. Cosh (c-0,06) 9. sinh (c-0) m 0. m sin (Ma-) Linar iffrntial Equations m m Ans.: C C C cosm C sin m cosm m. cosh (c-0) E : Solv 5 cos sin Solution :Th A.E. is 5 0, c c c cos sin 5 cos 5 sin 5 9 sin 5 cos 5 cos 5 sin 5 sin 5 5 sin 5( )( ) sin 5 sin 5( )( ) sin sin 0 5( ) Prof. M G Shrigan Pag 7
8 Enginring Mathmatics-III Linar iffrntial Equations Th gnral solution is givn b, sin sin sin 0 5( ) sin cos sin 0 50 c c c sin cos sin 0 50 E : Solv cos cos Solution : Th A.E. is c c cos c 0 0 i, sin c 0 c cos cos cos cos cos cos ( ) 0, cas fails ( ) cos cos cos ( ) cos 6 sin 6 cos cos cos 88 6 Prof. M G Shrigan Pag 8
9 Enginring Mathmatics-III Linar iffrntial Equations Th gnral Solution is givn b, c c cos c sin c c 88 cos sin 6 Rul No: (Ma-06)(c-0,0). ( 5) 5 (c-0,) C C 9 8. (Ma-05) Ans: C cos 5. 8 sin (Ma-06) Ans: C C 6. ( 6) (c-09) C sin 6 5 cos 7. 9 cos (Ma-09) C cos C sin 9 7 sin 6 8. sin 9. 6 (c-09) 50. ( ) (c-) Rul No: 5. ( ) sin (c-0,) Sin (c-06) Ans: C C 5 sin 0 cosh.cos Ans:C cos C sin 5 5. (c-06) 5. cos cosh (c-06) (Ma-0) 55. ( ) cos (Ma-06) (c-09,) sinh sin cosh cos 56. sin (Ma-05) Ans:C C cos sin 57. cos cosh (Ma-05) Prof. M G Shrigan Pag 9 0
10 Enginring Mathmatics-III 58. sinh (Ma-06) sin (Ma-05) Linar iffrntial Equations 60. cos (Ma-05) 6. sin (c-0) (c-0,) (Ma-) 6. cos (c-0) 6. sin (c-09) 65. ( 7 6) ( ) (c-) 66. sin (c-) 67. ( ) (Ma-0) 68. ( ) sin (Ma-0) (c-0) cos sin 0 Ans: c c Rul No: 5 Ans.: C C sin cos 69. Sin (Ma-05,07) (c-08,0) cos (Ma-09) 7. 9 sin 7. sin (c-05,08) 7. ( ) sin (c-) 7. sin (c-05)(ma-0,05) 75. sin (Ma-05) Ans: C Ans: C Ans: C cos C cos C cos C sin sin 6 sin cos sin cos cos sin 6 sin 76. sin (c-0,) 77. sin (Ma-0) Prof. M G Shrigan Pag 0
11 Enginring Mathmatics-III Linar iffrntial Equations sin (c-0) Ans: c c cos sin sin. Ans. c c sin sin cos 80. ( ) sin (Ma-) Solution: (c-0) B Variation of Paramtr Solv b variation of aramtr:. ( ) sin (c-09). 9 (c-08) sin cos Ans:C cos C sin sin 9 sin log sin 9 9. log (c-07) Ans: c c log Prof. M G Shrigan Pag
12 Enginring Mathmatics-III. ( ) cos c (c-0,0,) 5. (Ma-06,08) 6. (c-05,06)(ma-05) 7. ( ) sin (c-) 8. Cos (c-07) 9. ( ) tan (Ma-07,08) (c-09,,) 0. ( ) tan (Ma-07,) (c-). sin cos (Ma-89) (c-95). ( ) cot (c-05,06,) Linar iffrntial Equations Ans: c c log Ans: c c log log Ans: C C cos Ans: c c sin. ( 6 9) (Ma-09) Solution: b mtho of variation of aramtrs. (c-0) Th gnral solution is givn b, Prof. M G Shrigan Pag
13 Enginring Mathmatics-III Linar iffrntial Equations Cauch s iffrntial Equation. log (c-09). ( ) (Ma-08). log (c-07) Ans.: C C 6 log. log cos(log) (Ma-07) cos(log) sin(log) (Ma-05) Ans.: C C 5 log Ans.: C coslog C sinlog log log cos log (c-05) cos log C sin log coslog sinlog Ans.: C sin log sinlog 7. ( ) (Ma-05) Ans.: C C cos log C sinlog 8. ( 6) sin (c-0) log 5 50 Solution: This is Cauch s homognous quation, W us substitution (c 0) Lt ; From th givn quation w hav, Now, Prof. M G Shrigan Pag
14 Enginring Mathmatics-III Linar iffrntial Equations Changing an, w hav Lgnr s iffrntial Equation. ( ) ( ) sin[log( )] (Ma-0). ( ) ( ) sin[log( )] (c-07,09,). (c-09). ( ) ( ) cos[log( )] (Ma-) 5. ( ) ( ) 6 (c-05) 6. 6 (c-08) Ans:C C Prof. M G Shrigan Pag
15 Enginring Mathmatics-III 7. (Ma-09) (c-) Linar iffrntial Equations Ans.: C 8. sin[log( ) ] (Ma-)(c-07) 9. 6 (c-06) 0. (Ma-06) 5 C 6 log Ans.: C cos log C sinlog sinlog Ans.: C C 7. 8 (c-05) Ans.: C C. 8 coslog (c-0). ( ) ( ) 6 (Ma-07,,) (c-) Smmtric simultanous Equations z. (c-06) Ans : z C ; z C. z z z (c-0,) (Ma-) z. (Ma-0,) z. (c-) z z z 5. (c-09) Ans : C ; z C z( ) z( ) z 6. (Ma-05,06,07,08) (c-0) Ans : z C ;z C z z z Prof. M G Shrigan Pag 5
16 Enginring Mathmatics-III Linar iffrntial Equations z z z ( ) 7. (Ma-09) Ans: C ; z 8. (c-07) ( z ) (z ) z ( ) z z C 9. (c-05,0) (Ma-) Ans: C ; C 0. (c-0) (Ma-0) Ans : C ; z. (c-0) z z z. (Ma-05) z a z z z ( ) z z logz C z. (c-0) sin. z ( ) z (c-08,) Smmtric simultanous Equations. Solv th simultanous quations: a cost an a sin t (Ma-07)(c-0,0) a Ans: C cost C sin t sin t; C cost C sin t cost a u v. Solv th simultanous iffrntial quations: v sin ; u cos Givn at 0, u,v = 0. (c-09). Solv th sstm of quations: sin t ; cost Givn that whn t = 0, = 0, =. (Ma-) t t t t Ans : ; sin t (c-06). Solv th simultanous iffrntial quations: 0 ; 0 givn 0 an 00 at t 0. (c-0) Prof. M G Shrigan Pag 6
17 Enginring Mathmatics-III Linar iffrntial Equations 5. Solv for an : givn 0, 00 ; whn t 0 (Ma-05,09) 6. Solv : ( ) t ; ( ) t (c-) 7. Solv: 5 t ; 0 having bn givn that 0 at t 0 (c-0,07,0,) 8. Solv th simultanous quations: t ; 5 (c-) t 9. Solv simultanousl: 6 t & (c-08)(ma-05) Ans: C cost C t sin t t 9 0. Solv th sstm of quations: t ; t (c-09). Solv th simultanous quations: 5 z ; 8 z 5 (Ma-0) 5 t 7 ; t C C C C cost sin t t 9 5 t t. Solv th simultanous quations: ; 0 (Ma-06). Solv: 5 7 ; sin t (c-0) Ans.: C t C t. Solv th simultanous quations: ; sin t cost; 5. Th currnt an in th coul circuits ar givn b L R R( ) E ; t t C t t C 5 7sin t cost t t t t Ans: C C ; C C L R R( ) 0. Fin an, givn that 0 at t 0.(c-05,0,)(Ma-05) t Prof. M G Shrigan Pag 7
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