7-Mavzu: Mapleda differensial tenglamalarni yechish. Differensial tenglamalarni yechish funksiyalari. 1-, 2- va yuqori tartibli differensial tenglamalarni yechish

y''(0)=2 shartni (D @@ 2)( y)(0)  2 kabi berishga to’g’ri keladi yoki y'(1)=0 shart-ni: Eslatib o’tamiz, n-chi tartibli hosila (D@@n)( y) kabi yoziladi. Koshi masalasining yechimini topish: y(4)+y''=2cosx, y(0)=2, y'(0)=1, y''(0)=0, y'''(0)=0.

Fan: Kompyuter algebrasi tizimlari O’qituvchi: T.Djiyanov II-kurs 7-Mavzu..
D( y)(1)  0 .

• de:=diff(y(x),x$4)+diff(y(x),x$2)=2*cos(x);


 x2

 
   
 x4
 4  2
de : y( x)  y(x)  2 cos( x)

• cond:=y(0)=-2, D(y)(0)=1, (D@@2)(y)(0)=0, (D@@3)(y)(0)=0;

cond:=y(0)=2, D(y)(0)=1, (D(2))(y)(0)=0, (D(3))(y)(0)=0

• dsolve({de,cond},y(x));

y(x)=2cos(x)xsin(x)+x

 2 
 
Quyidagi chegaraviy masalaning yechimini topamiz: y'' y  2x   , y(0)  0 , y    0 . Yechim grafigini yasang.

• restart; de:=diff(y(x),x$2)+y(x)=2*x-Pi;





de:=

 x
y(x)  y(x)  2x  
2
 2

• cond:=y(0)=0,y(Pi/2)=0;

Fan: Kompyuter algebrasi tizimlari
O’qituvchi: T.Djiyanov
II-kurs
7-Mavzu..
cond : y(0)  0, y    0
 2 
 

• dsolve({de,cond},y(x));

y(x)=2x+cos(x)
Differensial tenglamalar sistemasi.
Differensial tenglamalar sistemasining (yoki Koshi masalasi-ning)
yechimini
dsolve komanda bilan topish mumkin, agarda unda quyidagilar ko’rsatilsa:
dsolve({sys},{x(t),y(t),…}), bu yerda sys  dif-ferensial tenglamalar sistemasi,
x(t),y(t),…  noaniq funk-siyalar ketma – ketligi.
Quyidagi differensial tenglamalar sistemasining yechimini toping:



,
3
et  1
 y  6x  3y 
2
et  1
x  4x  2 y 

• sys:=diff(x(t),t)=-4*x(t)-2*y(t)+2/(exp(t)-1),

diff(y(t),t)=6*x(t)+3*y(t)-3/(exp(t)-1):

• dsolve({sys},{x(t),y(t)});

{x(t)  3_ C1 4C1_ e(t)  2C2 _ 2C2 _ e(t)  2e(t) ln(et 1), y(t)  6 _ C1 6 _ C1et  3_ C2e(t)  4 _ C2  3e(t) ln(e(t) 1)}
_S1 va _S2 ixtiyoriy o’zgarmaslarga bog’liq bo’lgan x(t) va y(t) funk-siyalar topilgan.