Matematik tizimlarda tengsizliklar va tengsizliklar sistemasini yechish
![8Misol 2. Tenglamani yeching. 6,9 : 4,6 =
x : 5,4
> solve((6.9/4.6=x/5.4),x);
8.099999999
Misol 3.
x
3+ x
15 + x
35 + x
63 + x
99 + x
143 = 12 tenglamani yeching.
> solve((x/3+x/15+x/35+x/63+x/99+x/143)=12,x);
26
Misol 4.
3x−2
4 +2x+3
2 −2,5 x+2=0 tenglamani yeching.
> solve(((3*x-2)/4+(2*x+3)/2-2.5*x+2)=0,x);
4.
Agar tenglama bir necha yechimga ega bo`lsa, name buyrug’i orqali boshqa
hisoblash amallarini bajarishda yechimlarni tanlab olamiz. Berilgan tenglamaning
k- yechimini aniqlash uchun kvadrat qavslar ichida yechim tartibini ko`rsatish
lozim: name[k] . Masalan:
> x:=solve(x^2-a=0,x);
x:=−√a,√a
> x[1];
−√a
> x[2];
√a
> x[1]+x[2];
0
Misol 1. Tenglamani yeching.
1998 x2−2000 x+2=0
> solve(1998*x^2-2000*x+2=0,x);
1, 1
999
6](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_5.png?v=1)
![> x:=solve(1998*x^2-2000*x+2=0,x);x := 1, 1
999
> x[1];
1
> x[2];
1
999
Misol 2.
x1 va x2 sonlari 3x2−2x− 6=0 tenglamaning ildizlari bo’lsa, ildizlari
yig’indisi va ayirmasini toping.
> x:=solve(3*x^2-2*x-6=0,x);
x := 1
3 + 1
3 19 , 1
3 - 1
3 19
> x[1];
1
3 + 1
3 19
> x[2];
1
3 - 1
3 19
> x[1]+x[2];
2
3
> x[1]-x[2];
2
3 19
7](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_6.png?v=1)
![Maple dasturida yechish:
> s:=solve({x*x+y*y+x*y=8, x+y=3},{x,y});s := y = RootOf _Z 2 - 3 _Z + 1, label = _L1 ( ), x = -RootOf _Z 2 - 3 _Z + 1, label = _L1 ( ) + 3 { }
> assign(s); simplify(x*y);
1
Javob:
x⋅y =1
Misol 2. Sistemaning yechimini toping.
{x
2
+y
2
−2xy=1¿¿¿¿
Maple dasturida yechish:
> s:=solve({x^2+y^2-2*x*y=1, x+y=3},{x,y});
s := y = 1, x = 2 { }, y = 2, x = 1 { }
> s[1];
y = 1, x = 2 { }
> s[2];
y = 2, x = 1 { }
Javob: (2;1) va (1;2)
Misol 3. Ushbu
{x+y=3¿¿¿¿ tenglamalar sistemasidan x ni toping.
Maple dasturida yechish:
> s:=solve({x^2-y^2=6, x+y=3},{x,y});
s:={y= 1
2,x= 5
2}
> assign(s); simplify(x);
5
2
Javob: x=2,5
Misol 4. Tenglamalar sistemasini yeching.
{y+4=2¿¿¿¿
Maple dasturida yechish:
14](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_13.png?v=1)
![> s:=solve({y+4=2, (x^2)*y=-2},{x,y});s := y = -2, x = 1 { }, y = -2, x = -1 { }
> s[1];
y = -2, x = 1 { }
> s[2];
y = -2, x = -1 { }
Javob: (-1; -2), (1; -2)
Misol 4. Agar
x− y=5 va xy = 7 bo’lsa, x3y+xy 3 ning qiymati qancha bo’ladi?
Maple dasturida yechish:
> s:=solve({x-y=5, x*y=7},{x,y});
s := y = RootOf _Z 2 + 5 _Z - 7, label = _L1 ( ), x = RootOf _Z 2 + 5 _Z - 7, label = _L1 ( ) + 5 { }
> assign(s); simplify(x^3*y+x*y^3);
273
Javob:
x3y+xy 3 =273
Misol 5. Agar
a−b=12 va −ab +a2=144 bo’lsa, a ning qiymati qanchaga teng?
Maple dasturida yechish:
> k:=solve({a-b=12, (-a)*b+a^2=144}, {a,b});
k := a = 12 , b = 0 { }
> assign(k); simplify(a);
12
Javob: a=12
Misol 6. Agar
x2− 4xy +y2=4−2xy va x+y=12 bo’lsa, xy ning qiymatini toping.
Maple dasturida yechish:
> k:=solve({x^2-4*x*y+y^2=4-2*x*y, x+y=12}, {x,y});
k := y = 5, x = 7 { }, y = 7, x = 5 { }
> assign(k); simplify(x*y);
35
Javob:
xy = 35
15](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_14.png?v=1)
![bo`lganligi tufayli, Maple tizimi maxsus RootOf() funksiyasi yordamida
tenglamaning ixtiyoriy yechimini belgilaydi.
Misol:
> eq:=x^5+x^3+1=0; := eq x5 x3 1 0
> s:=solve(eq,x);
s ( ) RootOf , _Z 5 _Z 3 1 index 1 ( ) RootOf , _Z 5 _Z 3 1 index 2 , , :=
( ) RootOf , _Z 5 _Z 3 1 index 3 ( ) RootOf , _Z 5 _Z 3 1 index 4 , ,
( ) RootOf , _Z 5 _Z 3 1 index 5
> evalf(s[1]);
.6366631068 .6647015651 I
> solve(x=cos(x));
( ) RootOf _Z ( ) cos _Z
Oxirgi komandaning natijasi z-cos( z)=0 tenglamaning ixtiyoriy yechimini
ifodalaydi. _z belgi Maple tizimining hosil qilgan o`zgaruvchisi bo`lib, x ni
o`rniga almashtirilgan. Index parametri yechimning nomerini ko`rsatadi.
17](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_16.png?v=1)
![1.4. Trigonometrik va transendent tenglamalarni yechish usullari
Rekurrent va funksional tenglamalarni yechish
f butun funksiyaning eq tenglamasi uchun rsolve(eq,f) buyrug’I ishlatiladi.
f(n) funksiya uchun bir qancha boshlang’ich shartlar berilishi mumkin, bunda bu
rekurrent tenglama mos yechimga ega bo`ladi. Masalan :
> eq:=2*f(n)=3*f(n-1)-f(n-2); eq :=2f(n)=3f(n−1)−f(n−2)
> rsolve({eq,f(1)=0,f(2)=1},f);
2−4(
1
2)
n
solve universal buyrug’idan funksional tenglamalarni yechishda foydalaniladi.
Masalan:
> F:=solve(f(x)^2-3*f(x)+2*x,f);
F := proc ( x ) RootOf(_ Z ^2 - 3*_ Z + 2* x ) end
Tenglama natijada oshkormas ko`rinishni oladi. Maple bunday yechimlar bilan
ishlashi mumkin. Funksional tenglamaning oshkormas yechimi convert buyrug’I
yordamida elementar funksiyaga keltirilib topiladi. Yuqorida ko`rib o`tilgan
misolni oshkor ko`rinishda yechamiz:
> f:=convert(F(x),radical);
f:=3
2+1
2√9−8x
Trigonometrik tenglamalarni yechish
solve buyrug’i trigonometrik tenglamalarni yechish uchun qo`llanilib, [0;2 ]
intervaldagi bosh yechimni ko`rsatadi To`la yechimni ko`rsatish uchun
_ EnvAllSolutions := true davomiy buyrug’i qo`llaniladi.Masalan:
> _ EnvAllSolutions := true :
> solve ( sin ( x )= cos ( x ), x );
1
4π+πZ
~
21](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_20.png?v=1)
![Maple da π _ Z ~ belgisi butun tipning o ` zgarmasini aniqlaydi , bu berilgan
tenglama yechimi uchun
x:= π/4+πn , n – butun son .
Misol 1. Tenglamaning ( 0;2 ) oraliqqa tegishli yechimlarini toping.
cos x=− √2
2
Maple dasturida yechish:
> _EnvAllSolutions:=true:
> solve(cos(x)=-sqrt(2)/2,x);
3
4 p - 3
2 p _B 1~ + 2 p _Z2~
Misol 2.
sin 2x= (cos x−sin x)2 tenglamaning [ 0;2 ] kesmada nechta ildizi bor?
Maple dasturida yechish:
> _EnvAllSolutions:=true:
> solve(sin(2*x)=(cos(x)-sin(x))^2);
1
12 p + p _Z3~ , 5
12 p + p _Z3~
Transendent tenglamalarni yechish
Transendent tenglamani yechishda, yechim aniq ko`rinishda bo`lishi uchun
solve buyrug’idan avval _ EnvExplicit := true buyrug ’ ini ifodalash kerak .
> eq:={ 7*3^x-3*2^(z+y-x+2)=15, 2*3^(x+1)+
3*2^(z+y-x)=66, ln(x+y+z)-3*ln(x)-ln(y*z)=-ln(4) }:
> _EnvExplicit:=true:
> s:=solve(eq,{x,y,z}):
> simplify(s[1]);simplify(s[2]);
{ x =2, y =3, z =1}, { x =2, y =1, z =3}
Quyida keltirilgan ko’rsatkichli tenglama va tengsizliklarning yechimi
Maple muhitida topilgan.
Misol 1. Tenglamani yeching.
4x−4=0,5
Maple dasturida yechish:
> eq:={4^(x-4)=0.5};
22](https://docx.uz/documents/940e2543-f798-4e72-b4ba-b6acd828364e/page_21.png?v=1)
Matematik tizimlarda tengsizliklar va tengsizliklar sistemasini yechish