O'rta maxsus ta'lim

y = Cx2 parabolalaroilasiningortogonaltrayektoriyalarini toping.

r^ a( 14cosip) kardioidalar oilasining ortogonal trayektoriyalarini toping.

5 .2 \.y^Cx to’g ’ri chiziqlar oilasining izogonal trayektoriyalarini toping

22. 
    x 2 = 2a(y - W 3 ) egri chiziqlarni 60° burchak ostida kesuvchi izogonal trayektoriyalar oilasini toping.
    
23. 
    у 2 = 4Cx parabolalar oilasining izogonal trayektoriyalarini toping. Kesishish burchagi 45° ga teng.
    
24. 
    r2=a2cos2
    lemniskatalar oilasining ortogonal trayektoriyalarini toping.
    

I - bohga doir nmol vm magatahirnmg javobfain ,-1

l . I . arctgx + arctgy = С . 1.2. I + y 2= Cx2 .1 .3 . Vl + дг + -J] + у 1 = С .

1.4. - ^{1 I}
y - 2 2(jc + 1)
-С. 1.5. y = s in ( C In ( l + .x2)); y = l . 1.6. y = f a x - 3 x 2 +C

1 . 7 . 2 y - 2 a r c rg y - 3 1 n } x - l | + ln |;c + l |= C . 1.8. y - x ( \ n \ y \ + \) =Cycosx, y= 0 . l^ ./g ^ jt + s i n ^ = C . l.lO .x + y ^ l n ^ j c + lXy + l ) ) , . ) ^ - 1. l . l l . x - y + ln\xy\ =C , у=0АЛ2. ( x - l ) 2+ y 2 = C2. 1.13.cosy = C e o s jt.1.14.(1+ e >)eI =C .

1.15. y = Ce^ . 1.16. y = С(дг2—4 ) . 1.17. y = C co sx . 1.18. y =C(x +J x 2 +a2).
^{1.19. In x +у} = С , 0. 1.20. InJjcyJ +xy~ C . 1.21. x +у - 0 . 1.22. 2ey = e* + 1.
ХУ

1.23. х 2+у 2 = 2

/
1+ In ^X

. 1.24. у = еГ' г. 1.25. y = 2 sin 2x — .

\ ^У ) ²
1.26. ~ J y - x l n x - x + 1. 1.27. x 2= 2 + 2у 2. 1.28. sinx. 1.29. 5 min 56s.
1.30. — -— j. 1.31.y=-2e3\ 1.32. y =~ . 1.33. 60 min.
40 In2,5 x
2 x
2 . 1. tg— = InlCxl. 1.35.Jt = Ce>*' .2 .2 . y 2 + 3 x y - 2 x 2 = C .
X
2.3. jc= C ( ln y - l n x - 1) 2.4. x = C‘$ ( y / x ) - 2.5. y =S x ~-2l ~ ~ -
J l +tg H y/x) ' 2(x - 1)
2 .6 . Зу + x - ln(x - 2 y) = С . 2 .7 .y 1-3xy+2x2- С. 2.8. x 2-y2 = C * .
у 2 _J.
2.9. In Cx =- е ~ж. 2.10. y = — — . 2.11. y 2 = x 2(l + C x). 2.12. y = xe г“
C +x
2 .13 . y =x e 2 . 2.14. sin— + ln|x| = 0 . 2.15. y = - x . 2.16. J x 2 +y 2 =ex
x
2Л7. y = 4e~**~ . 2.18. y 1 = 2C ^x + ^-j. 2.19. y = l ™ . 2.20. y = ^ -

3.1. y = - i + O r ' 2 . 3.2. y = ( x 2+ C ) e ' \ 3.3. y = (x3 + C ) ln x .

3.4. x + —y 1+ —y + —= Ce2y. 3.5. y = ( 6 + C ) s in x . 3.6. y = ?-

2 2 4 V:
3.7. у 2= x ( C - ln x ) . 3.8. y - x V = C . 3.9. x 2+ / = Cy2. 3.10. y = _{C - e smx}
3.11. >, = 4 x + ^ g j tl £ !) + C’- 3.12. y = 3.13. y = - 1
л / ^ Т х 1 ' ' 2x 2 xln C x
3.14. y = lnx + — .3 .15 . у = -^—!-+ - T ^ - 3.16. у = ± ; --.
X 3 7 2 ^ ' 4 ^ 7
2
3.17. y = x - x 2. 3.18. 3.19. у = - ^ — . 3.20. y = l . 3.21. y 3 = x - 2 e ' *.
C O S X cos*
3.22. y = — j=L =— . 3.23. v = (v0+ 6> +*(a =~ , Ь=Ц ^
V l- x 2 + l 2w *,
3.24. / +Rsinrot-соI. cosa>t), (zanjirdagi kuchlanish L— +R!


qonuniyat bilan o ’zgaradi). 3.25. v = — f / - —+ —e *"’ 1
к \ к к j
3.26. у = 2( lT a 2) + j - (tenglamasi | j y - x 2y j = o! ). 3.27. y = 2 x - j c l n | x |

4 .1 .(x+ 7 )(jc-.y)V =C . 4.2. six2- у 2 - x = C . 4.3. (| + е'^ = С .
4.4. x J ry ’-x2-xy ty 2~C. 4.5. x3y->x2-y=Cxy. 4.6. xe* \ ye +3x-2_y = C 4.7. x 2 + ^ + e " = С . 4.8. x3 + 3d2/ + d4 = С . 4.9. xe" - у 2 = С .
4.10. x 2+ у 2- 2arctg— =С . 4.11. x V - y =C . 4.12. x 2cos2> + .y = C .
x
4.13. — C| —^ + x = C . 4.14. 6x 2 -t- Sxy +y 2 - 9 x - 4 y = C .
4.15.-^x-²-^——^y2 + yx =C; fi =y. 4.16. jc——^v = ¹r.
2 л *
4.17. y 3+ x 3( ln x - l) = Cx2; p = —¹r 4.18. x 2 ---⁷ Зх_у = C; p = —¹ .
* > >
5.1. / = 2x + C ;y = - ix 2+C '. 5.2. y = C;y =±- Jx+C .


5.3. >=£ ; j , =£ . 5.4. J, = ±£- +C; j/ =( V
лг jc 2
5.5. у = 0 ; Г =О_y ,₌+_p1_2eУ_pЧ С . 5-6.1_{[y = p} *₊=_coto_sp' ₊+_psi_sn_m? _{p +C} .


S 3 . , . X I ХШ* “ ’ * С . S . » . , . C ^ a ^ C ’ . y . - U * “ )
y = psinp +cosp 4

5.9. ^ = Cx + V ^ , x 2 + / = l 5.10 \* =W - P ) +C e ' J u ^

Iy = 2(p2-] ) 2+Ce"(l + p ) + p 2
x = p - l n p + C , ,
5.11. •{ ( p - 1 )2 . 5.12. y = Cx +— , y =- x 2. 5.13. y = 4e2, y = -4 e 2.
y =xp^z - p ⁴
5.14. У =~ - 5.15. (y-x-2 a)2=&ax. 5.16. у 2=Cx~m + ~ ~ . 5 .17 . xy^a2.
2 2
5.18. S=at2, a-o’zgarmas son. 5.19. ^ - + -^- = C 2. 5.20. p =Csm2^ .
_L -v
5.21. x 2 + у 2 = Се* , k = tga . 5.22. y 2 - C(x - y 4 3 ).
6 2y - x
52Ъ. y 1 - x y + 2 x 2 =Ce^ . 5 . 2 4 . /-2 = C s i n 2 < o .

1 BOB. YUQORI TARTIBLI DIFFERENSIAL TENGLAMALAR.

!-§• Tartibini paanytirish mumkin bo4g«n diffcrenstol tenglam »br. F '

n-tartibli differensial tenglamani simvolik ravishda


F (x,y,y ..... У"'1’, / V O (1)

ko’rinishda yoki bu tenglamani n-tartibli hosilaga nisbatan yechib bo’lsa,

у <п)=Лх,у,у..... /" '> ) (2)


ko ’rihishda yozish mumkin.
и-tartibli differensial tenglamaning umumiy yechimi x ga va n-ta ixtiyoriy o ’zgarmaslarga bog’liq bo’ladi: у = g ( x , C l,C 2,...,C „).
Shu sababli umumiy yechimdan xususiy yechimni ajratib olish uchun ixtiyoriy o ’zgarmaslami aniqlashga imkon beradigan ba’zi qo’shimcha shartlar ham berilgan bo’lishi kerak. Bu shartlami izlanayotgan funksiyaning va uning (n-1 (-tartibgacha (y ham kiradi) barcha hosilalaming biror nuqtadagi qiymatlarini, ya’ni x=xo da


У(х0) =у 0, y'(xu) = y i,...,y t ’-l)(xa) = y n ] (3)

berish bilan hosil qilish mumkin. (3) sistema boshlang'ich shartlar sistemasi deyiladi. Berilgan (2) differensial tenglamaning (3) boshlang’ich shartlar sistemasini qanoatlantiruchi xususiy yechimini topish masalasi Koshi masalasi deyiladi.

Yuqori tartibli differensial tenglamalami integrallash masalasi birinchi tartibli tenglamani integrallash masalasidan ancha qiyin bo’lib, har doim ham birinchi tartibli tenglamani integrallashga keltiraverilmaydi. Shunday bo’lsada chiziqli tenglamalardan tashqari barcha turdagi yuqori tartibli tenglamalar uchun integrallashning asosiy usuli tartibini pasaytirish, ya’ni berilgan tenglamani unda o ’zgaruvchilami almashtirish orqali tartibi pastroq tenglamaga keltirish bo’lib hisoblanadi. Biroq tenglamaning tartibini pasaytirishga har doim ham erishish mumkin emas. Biz bu yerda tenglama tartibini pasaytirishga imkon beradigan n-tartibli tengtamalarning eng sodda turlari bilan tanishamiz.

1. 
   Ushbu