PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II DIAFORIKES EXISWSEIS.

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PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II DIAFORIKES EXISWSEIS h Seirˆ Ask sewn Diaforikèc eis seic <<qwrizomènwn metablht n>> << OmogeneÐc >> diaforikèc eis seic Diaforikèc eis seic <<qwrizomènwn metablht n>> Jèma. Na brejeð h genik lôsh twn parakˆtw D.E.: i. y = y 2 cos ii. y = y 2 + ) iii. y = y iv. 2 d + cos y = 0 v. y 2 + d = 0. Jèma 2. Na breðte th lôsh thc diaforik c eðswshc y 2 + ) d + 2 y y ) = 0 pou epalhjeôei tic arqikèc sunj kec y = ìtan = 0. Na lujoôn oi D.E.: i. y d + + ) = 0 ii. y = y y iii. y = e y+e y iv. e d y = 0, y0) =. ASKHSEIS PROS LUSH

<< OmogeneÐc >> diaforikèc eis seic Jèma 3. Na brejeð h genik lôsh twn diaforik n eis sewn : a) 2 + y 2) d 2y = 0 b) y = y g) y = 3 +y 3 y 2. [ + ln y )] Jèma 4. Na lujeð h diaforik eðswsh 2yy y 2 + 2 = 0. Jèma 5. Na lujeð h diaforik eðswsh y = + y y. ASKHSEIS PROS LUSH DeÐte ìti oi parakˆtw D.E. eðnai omogeneðc kai lôste tic: i. y = y + y ii. 2 + 2y ) d + y = 0 iii. y = y 2 y 2 iv. 2 + y + y 2) d 2 = 0 v. y d + 2 y 2) = 0 vi. y = 2 +3y 2 2y vii. y = +3y y. Anaplhrwt c Kajhght c : Dr. Pappˆc G. Alèandroc 2

Jèma. Na brejeð h genik lôsh twn parakˆtw D.E.: i. y = y 2 cos ii. y = y 2 + ) iii. y = y iv. 2 d + cos y = 0 v. y 2 + d = 0. LUSH i. H D.E. èqei th morf me y = y 2 cos y = f) gy) f) = cos kai gy) = y 2. MporeÐ na grafeð sth morf Prˆgmati Me olokl rwsh thc ) paðrnw P) d + Qy) = 0 y = y 2 cos d = y2 cos = y 2 cos d = cos d y2 = cos d ) y2 y 2 cos d = c pou eðnai h lôsh thc D.E. se peplegmènh morf. 3 sin = c to c eðnai aujaðreth stajerˆ) y

ii. H D.E. èqei th morf me y = y 2 + ) y = f) gy) f) = kai gy) = y 2 +. MporeÐ na grafeð sth morf P) d + Qy) = 0 Prˆgmati y = y 2 + ) Me olokl rwsh thc ) paðrnw d = y 2 + ) = y 2 + ) d y 2 + = d y 2 d = 0 ) + y 2 + y 2 + d = c d = c τoξεϕy 2 2 = c. H teleutaða sqèsh apoteleð th genik lôsh thc dosmènhc D.E. se peplegmènh morf. 4

iii. H D.E. y = y èqei th morf me y = y y = f) gy) f) = kai gy) = y. MporeÐ na grafeð sth morf P) d + Qy) = 0 Prˆgmati Me olokl rwsh thc ) paðrnw ìpou y = y d = y = y d y = d y d = 0 ) d y = 0 ln y ln = c ln y = c y = e c y = c y = c 2 c = e c kai c 2 = ±c. 5

iv. 2 d + cos y = 0. Me olokl rwsh thc D.E. paðrnw: 2 d + cos y = c. 3 3 + sin y = c pou eðnai h genik lôsh thc dosmènhc D.E. se peplegmènh morf. v. y 2 + d = 0 H dosmènh D.E. mporeð na grafeð: Me olokl rwsh prokôptei: y 2 + d = 2 d = y 2 + ) 2 d y 2 + ) = 0 2 d y 2 + ) = c 3 3 y 2 = c 3 3 3 y3 y = c. 6

Jèma 2. Na breðte th lôsh thc diaforik c eðswshc y 2 + ) d + 2 y y ) = 0 pou epalhjeôei tic arqikèc sunj kec y = ìtan = 0. LUSH H D.E. y 2 + ) d + 2 y y ) = 0 mporeð na grafeð sth morf Prˆgmati Me olokl rwsh prokôptei: ìpou P) d + Qy) = 0 y 2 + ) d + y 2 ) = 0 2 d + y + y 2 = 0 2 d + y + y 2 = c 2 ln 2 + 2 ln + y 2) = c ln 2 + y 2) = 2c + y 2 ) 2 = e c = c 2 c = 2c kai c 2 = e c. H lôsh thc D.E. se peplegmènh morf eðnai: LÔseic eðnai kai oi eujeðec + y 2 = c 2 2. = kai =. Gia tic arqikèc sunj kec èqoume: 'Wste h zhtoômenh merik lôsh eðnai: + 2 = + y 2 = c 2 = 2. c 2 0 2 2 2. 7

Jèma 3. Na brejeð h genik lôsh twn diaforik n eis sewn: a) 2 + y 2) d 2y = 0 b) y = y g) y = 3 +y 3 y 2. [ + ln y )] LUSH a) H dosmènh D.E. 2 + y 2) d 2y = 0 èqei th morf ìpou P, y) d + Q, y) = 0, P, y) = 2 + y 2 kai Q, y) = 2y eðnai omogeneðc sunart seic Ðdiou bajmoô, afoô isqôoun: P λ, λy) = λ) 2 + λy) 2 = λ 2 2 + y 2) = λ 2 P, y) k = 2 Qλ, λy) = 2λ λy = λ 2 2y) = λ 2 Q, y) k = 2. 'Ara èqoume na lôsoume mða omogen D.E. Je roume thn antikatˆstash: Apì th sqèsh ) paðrnw: y = u [u = u)] ) = du ) = u ) d = u + u ) d = u + u) d 2) H dosmènh D.E. grˆfetai: 2 + u 2 2) d 2 u u + u) d = 0 2 + u 2) ) du d 2 2 u d + u d = 0 2 [ + u 2) d 2u du 2u 2 d ] = 0 8

kai apì to mhdenismì thc agkôlhc paðrnoume: u 2 ) d 2u du = 0 dhlad D.E. qwrizomènwn metablht n. An oloklhr soume paðrnoume: 2 [ u 2) d 2u du ] = 0 3) u 2 ) d = 2u du d = ln = ln = d = 2udu u 2 4) 2udu u 2 u 2 du2 u 2 d u 2 ) ln = ln u 2 + c ln + ln u 2 = c ln u 2 ) = c u 2 ) = c 5) ìtan o logˆrijmoc mðac posìthtac eðnai stajerìc, tìte h posìthta eðnai stajer : ln z = c z = e c ). Me antikatˆstash sth sqèsh 5) tou u = y prokôptei h genik lôsh thc dosmènhc D.E. : se peplegmènh morf. [ y ) ] 2 9 y 2 = c = c

b) H dosmènh D.E. grˆfetai isodônama: y = y [ y + ln )] d = y [ y + ln )] [ y = y + ln d )] [ y y + ln d = 0 ) )] Oi sunart seic [ y P, y) = y + ln )] kai Q, y) = eðnai omogeneðc pr tou bajmoô omogèneiac, afoô isqôoun : P λ, λy) = λy ˆra h D.E. ) eðnai omogen c. [ )] λy + ln λ [ y = λ y + ln = λ P, y) k = )] Qλ, λy) = λ = λ ) = λ Q, y) k = Je roume thn antikatˆstash: Apì th sqèsh ) paðrnw: kai h diaforik eðswsh grˆfetai: y = u [u = u)] ) = du ) = u ) d = u + u ) d = u + u) d = u d + u d 2) [ u )] u + ln d u d + u d) = 0 [u d + u ln u d u d u d] = 0. Me u d = du 0

kai mhdenismì thc agkôlhc paðrnoume: dhlad D.E. qwrizomènwn metablht n. u ln ud du = 0 du u ln u = d An oloklhr soume paðrnoume: du d u ln u = dln u) ln u = d ln ln u = ln + c ln ln u = c ln u = c kai me antikatˆstash u = y, ftˆnoume sth genik lôsh thc dosmènhc D.E. se peplegmènh morf : y ) ln = c. Parat rhsh : 'Otan se D.E. a' tˆhc upˆrqei o lìgoc y y eðnai omogen c.

g) H dosmènh D.E. grˆfetai: y = 3 + y 3 y 2 d = 3 + y 3 y 2 3 + y 3) d y 2 = 0 ) H ) eðnai profan c mða omogen c D.E. Jewr thn antikatˆstash: y = u = u + u)d H ) grˆfetai: 3 + u 3 3) d u 2 2 u + u)d = 0 3 + u 3) ) du d 3 u 2 d + u d = 0 3 [ + u 3 u 3) d u 2 du ] = 0 3 d u 2 du ) = 0. Apì to mhdenismì thc parènjeshc prokôptei: d u 2 du = 0 d u2 du = 0. H teleutaða D.E. eðnai D.E. qwrizomènwn metablht n. An oloklhr soume paðrnoume: An jèsoume sthn 2) to ln 3 u3 = c. 2) u = y prokôptei: ln 3 y ) 3 = c pou eðnai h genik lôsh thc D.E. se peplegmènh morf. 2

Jèma 4. Na lujeð h diaforik eðswsh 2yy y 2 + 2 = 0. LUSH H dosmènh D.E. grˆfetai isodônama: 2yy y 2 + 2 = 0 2y d y2 + 2 = 0 2y d = y2 2 2y = y 2 2) d y 2 2) d 2y = 0 ) Oi sunart seic P, y) = y 2 2) kai Q, y) = 2y eðnai omogeneðc sunart seic Ðdiou bajmoô, afoô isqôoun: P λ, λy) = [ λy) 2 λ) 2] = λ 2 y 2 λ 2 2) = λ 2 y 2 2) = λ 2 P, y) k = 2 Qλ, λy) = 2λ)λy) = λ 2 2y) = λ Q, y) k = 2 ˆra h D.E. ) eðnai omogen c. Je roume thn antikatˆstash: Apì th sqèsh ) paðrnw: y = u [u = u)] ) = du ) = u ) d = u + u ) d = u + u) d 2) kai h diaforik eðswsh grˆfetai: u 2 2 2) d 2u u + u) d = 0 3

2 u 2 ) d 2 2 u u + u) d = 0 2 [ u 2 ) d 2u u + u) d ] = 0 2 0 u 2 ) d 2u u + u) d = 0 u 2 ) ) du d 2u d + u d = 0 u 2 ) d 2u du + ud) = 0 u 2 ) d 2udu 2u 2 d = 0 u 2 ) d 2udu = 0 u 2 + ) d 2udu = 0 u 2 + ) d = 2udu d = 2u u 2 + du d + 2u u 2 + du = 0 H teleutaða D.E. eðnai D.E. qwrizomènwn metablht n. An oloklhr soume paðrnoume: d + 2u u 2 + du = c ln + u 2 + d u 2 + ) = c ln + ln u 2 + ) = c ln u 2 + ) = c 4

ìtan o logˆrijmoc mðac posìthtac eðnai stajerìc, tìte h posìthta eðnai stajer ) u 2 + ) = c An jèsoume u = y prokôptei: ) y 2 2 + = c pou eðnai h genik lôsh thc D.E. se peplegmènh morf. 5

Jèma 5. Na lujeð h diaforik eðswsh y = + y y. LUSH H dosmènh D.E. grˆfetai isodônama: y = + y y d = + y y y) = + y)d + y)d y) = 0 ) Oi sunart seic P, y) = + y) kai Q, y) = y) eðnai omogeneðc pr tou bajmoô omogèneiac, afoô isqôoun : ˆra h D.E. ) eðnai omogen c. P λ, λy) = λ + λy) = λ + y) = λ P, y) k = Qλ, λy) = λ λy) = λ [ y)] = λ Q, y) k = Je roume thn antikatˆstash: y = u [u = u)] ) Apì th sqèsh ) paðrnw: = du ) = u ) d = u + u ) d kai h diaforik eðswsh grˆfetai: = u + u) d 2) + u)d u) u + u) d = 0 + u)d u) u + u) d = 0 6

[ + u)d u) u + u) d] = 0 0 + u)d u) u + u) d = 0 ) du + u)d u) d + u d = 0 + u)d u) du + ud) = 0 d + ud du ud + udu + u 2 d = 0 + u 2 ) d + u )du = 0 + u 2 ) d = u )du d = u + u 2 du d + u + u 2 du = 0 H teleutaða D.E. eðnai D.E. qwrizomènwn metablht n. An oloklhr soume paðrnoume: An jèsoume prokôptei: u d + + u 2 du = c u ln + u 2 + du u 2 + du = c ln + 2 u 2 + d u 2 + ) τoξεϕu = c ln + 2 ln u 2 + ) τoξεϕu = c ln + 2 ln y 2 u = y ) 2 + y τoξεϕ = c ) pou eðnai h genik lôsh thc D.E. se peplegmènh morf. 7