*Agbo, F. I. and # Olowu, O.O. *Department of Production Engineering, University of Benin, Benin City, Nigeria.

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1 REDUCING REDUCIBLE LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH FUNCTION COEFFICIENTS TO LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ABSTRACT *Abo, F. I. an # Olowu, O.O. *Department o Prouction Enineerin, Universit o Benin, Benin Cit, Nieria. #Department o Mathematics, Universit o Benin, Benin Cit, Nieria. In this article, we propose a eneralize metho or obtainin a substitution or reucin a reucible linear orinar ierential equation with unction coeicients (RLDEF) to a linear orinar ierential equation with constant coeicient (LDE). This propose metho was also use to obtain the alrea known substitutions or the Euler s an Leenre s homoeneous secon orer linear ierential equation. The erive metho is able to reuce quite a lare number o RLDEF to LDE incluin the Euler s an Leenre s homoeneous secon orer linear ierential equation. However, these RLDEF (homoeneous an inhomoeneous) must satis the conition or reucibilit, which is also propose beore the substitution is erive. the conition or reucibilit is base on the orer o the ierential equation. In this article, the conition or reucibilit is presente or a secon an thir orer LDEF. Kewors: reucibilit, eneralize, ierential equations..0 Introuction The solutions to RLDEF have been one o the major problems in solvin linear orinar ierential equations (ODEs). These RLDEF can be solve either b substitution which transorms it into a LDE or b knowin one o the solutions o the RLDEF. However, reucin these RLDEF b substitution to a LDE requires ettin the riht substitution. These substitutions are majorl otten b trial an error metho especiall when it is not the Euler s or Leenre s homoeneous ierential equation orm which has stanar substitutions [, ]. This paper proposes a stanar metho o eterminin i a linear orinar ierential equation with unction coeicients is reucible an subsequentl a metho on how to etermine the substitution that will reuce it i reucible to LDE, which hitherto oes not eist in literature to the best o our knowlee. #corresponin author: ohenewaire.olowu@uniben.eu

2 .0 Proposition The homoeneous secon orer orinar ierential equation with unction coeicient a 0, where a is a constant, is reucible i ' b hols an it is reucible to a ierential equation with constant coeicient b the substitution z Proo. Given a secon orer ierential equation o the orm ) ( ) ( a 0.0 Where a is a constant Consier the substitution z r(). ( ) ( ) ( ) ( ) ( ). r"( ) ( ) ( ) r"( ) ( ) Substitute (.) an (.) into (.0) an simpli ives ( ) ( ). ( ) ( ) r"( ) ( ) ( ) a 0.

3 ( ) r"( ) ( ) ( ) b. Where b is a constant ( ). 6 So that we have b From (.6) a 0.7 ( ). 8 ( ) r.9 r" ( ) '( ) ( ).0 Substitute (.8) an (.0) into (.) an ater simpliication we have ( ) '( ) b ( ). Also rom (.8) an (.), we have z r. (.0) is reucible i (.) hols an (.) is a substitution require to reuce (.0) to a LDE.0 Proposition

4 The homoeneous thir orer orinar ierential equation with unction coeicient where a is a constant, is reucible i p a 0 () '() b an p '' ' b ' c hol an it is 9 reucible to a ierential equation with constant coeicient b the substitution z. Proo Consier a thir orer orinar ierential equation o the orm: p a 0. 0 Where a is a constant Let z r(),. So, ( ) ( ) p() p() (). r"( ) () ()r"() () (). r '''( ) ' ''() '.

5 Substitute (.), (.) an (.) into (.0) an ater simpliication ives p() () ()r"() () () ' ''() a 0 () () ' b. p() () ()r"() ''() c. 6 Where b an c are constants (). 7 So that we have b c a From (.7) 0.8 ( ).9 r.0 r"() ''() '() ''() 9 7. '. Substitute (.9) an (.) into (.) an simpli () '() b. Substitute (.9), (.) an (.) an simpli

6 ' ' p '' b c 9 Also rom (.) an (.0), we have. z r..0 Proposition The homoeneous thir orer orinar ierential equation with unction coeicient p h a 0 () '() b, p '' ' 6 where a is a constant, is reucible i b ' c, ' ' '' ' c ' h ''' ' '' b hol an it is reucible to a ierential equation with constant coeicient b the substitution z. an Proo Consier a ourth orer orinar ierential equation o the orm: p h a 0. 0 Where a is a constant Let z r(),. So, 6

7 ( ) ( ) h h. r" pr" p (). p '' ' ''() '. ''' '' ' 6 ' ''' '' ' 6 '. Substitute (.), (.), (.) an (.) into (.0) an ater simpliication ives () () 6 ' p () ' '' ' h p '() () '' ''' a

8 () () 6 ' b. 7 p () ' '' ' c. 8 h p '() () '' '''. 9 Where b, c, are constants (). 0 So that we have b c a 0. From (.7) (). r. r"() '(). ''() ''() 6 9 '. 8

9 '''() 6 '''() 6 ' '' 8 '. 6 9 ' '' Substitute (.) an (.) into (.7) an simpli () '() b. 7 Substitute (.), (.) an (.) into (.8) an simpli p '' ' 6 b ' c. 8 Substitute (.), (.), (.) an (.6) into (.9) an simpli h c ' '' ' 6 ' 6 ' ' 6 '' b '. 9 '' Also rom (.) an (.), we have z r.0 Corollar In eneral we conclue that iven an RLDEF o the orm... a 0 where a is a constant, is reucible to a LDE b the substitution.0 Illustrative eamples n n n n n, n z n n 9

10 . Solve the ierential equation with alebraic coeicient SOLUTION First we check i it is reucible, B (.), we have b. Where b 7 Which satisies in (.0) an hence (.0) is reucible, To reuce (.0) we appl (.) to obtain z tan. From (.), we have Substitutin (.) an (.) into (.0) an simpli Ae Ae Be z z tan tan Be

11 . Solve the ierential equation with loarithmic coeicient ( ln ) (ln ) 0.8 SOLUTION To check i the equation is reucible, From (.), ln ln ln b Where b Which satisies in (.8) an hence (.8) is reucible, B (.), we reuce (.8) b the substitution z() ln(ln ).9 (ln ) (ln ).0 ln ln. Substitutin (.0) an (.) into (.8) an simpli, we have 0. A B ln. ln. Solve the ierential equation with trionometr coeicient cos sin SOLUTION cos 8cos.

12 cos sin cos Divie throuh b cos 8cos sec tan sec 8 cos To check i the equation is reucible, From (.), sec sec tan bsec Where b 0 Which satisies in (.) an hence (.) is reucible To reuce (.) we appl (.) to obtain z sin tan sec tan sec..6 sec tan sec.7 Substitutin (.6) an (.7) into (.) an simpli, we have 8 z A cos sin B sin sin sin. 8. Solve the ierential equation with trionometr coeicient 8 cos ec cos ec cot 0.9 SOLUTION

13 To check i the equation is reucible, From (.), cos ec cos ec cot b cos ec Where b 0 Which satisies in (.9) an hence (.9) is reucible, B (.), we reuce (.9) b the substitution z sin.0 cos ec cot cot cos ec. cos ec cos ec cot. Substitutin (.) an (.) into (.9) an simpli, we have 0. A cos sin B sin sin.. Solve the thir orer linear ierential equation with alebraic coeicients 9 0. SOLUTION 9 0,, p 9

14 I the equation is reucible then rom (.) an (.) () b an p b c 9 Where b 0 an a 0, an p 9 Implies (.) is reucible, usin (.) to et the substitution z.6 B substitutin (.6) into (.) an simpli we have 0 Ae z B cos z z C sin. 7 Put (.6) into (.7) Ae z B cos C sin. 8.6 Solve the ourth orer ierential equation with alebraic coeicient SOLUTION Compare (.9) an (.0), we have,, p, h 70 6 I the equation is reucible then rom (.7), (.8) an (.9) b, p b c

15 h b c Where b 0, c 0, 0 Implies (.9) is reucible, usin (.0) to et the substitution z.0 D 8 0 D, i A cosh z B sinh z C cos z D sin z A cosh B sinh C cos D sin 6.0 Derivation o the substitution or the Cauch-Euler s homoeneous ierential equation usin the propose metho. The Cauch-Euler s homoeneous ierential equation is o the orm a a where an a (6.0) is reucible i ' b b Where b a The substitution is z

16 z ln 6. Substitutin (6.) into (6.0) we have, (a ) a (6.) is the substitution iven b Cauch-Euler to reuce (6.0) to a ierential equation with constant coeicient as seen in (6.). 7.0 Derivin the substitution or the Leenre s homoeneous ierential equation usin the propose metho. The Leenre s homoeneous ierential equation is o the orm a c) (a c) a 0 ( where (a c) an (a c) (7.0) is reucible i ' b a ba c Where b a The substitution is z a c a c 7. z ln Substitutin (7.) into (7.0) we have, ( a ) a (7.) is the substitution propose b Leenre to reuce (7.0) to a ierential equation with constant coeicient as seen in (7.). 6

17 8.0 CONCLUTION The propose metho has able to reuce linear orinar ierential equation with unction coeicients to linear orinar ierential equation with constant coeicient as illustrate in section three. Section, orinar ierential equation with alebraic, loarithmic an trionometr coeicients were treate. The Cauch-Euler s an Leenre substitution or reucin equations o speciic orm were erive usin our propose metho. The erive substitution is the same as the ones propose b Cauch-Euler an Leenre to solve their orm o equations as seen sections our an ive. The propose metho can be use to solve problems that even the popular Frobenius metho will be unable to solve. Reerences [] Stevenson, G. (97), mathematical methos or science stuents, Lonman, Lonon an New York. [] H. K. DASS (0), Avance enineerin mathematics, S. Chan, New Delhi. [] 7