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7

MAPLE.

: Maple.

 

. Maple :

 

diff (a, x1, x2,, xn), diff(a, [x1, x2, , xn]),

 

 

Diff (a, x1, x2,, xn), Diff(a, [x1, x2, , xn]).

, f(x1, x2,,xn) , .

diff(f(x),x) f(x) x. n, 1, , , Diff(diff(f(x),x),y. x$n, n .,diff(f(x),x$4), x, diff(f(x),x,x,x,x).

. simplify factor expand, , .

, D(f), f .

 

 

. Maple :

 

int(f, x), int(f, x=a..b),

int(f, x = a..b, continuous);

 

 

Int(f, x), Int(f, x=a..b),

Int(f, x=a..b, continuous).

 

f ,

x , ,

b ,

continuous . , Maple . .

evalf(int(f, x=a..b),), e ( ).

, infinity.

, , - . - , . assume(expr1), expr1 . additionally(expr2), expr2 , .

Maple (~), , a, , : a~.

a about(a).

 

Maple dsolve(eq,var,options ), eq , var , options . , , : type=exact. diff, , y'' + y = x diff(y(x),x$2)+y(x)=x.

, . Maple , , _ 1, _ 2, ..

dsolve . , (, ), rhs(%).

, , , dsolve , , y''(0)=2 y '(1)=0: . , n - .

dsolve ( ), : dsolve({sys},{x(t),y(t),}), sys - , x(t),y(t), - .

, Maple Detools.

DEplot DEtools . . : DEplot: DEplot(de, vars, range, x=1..2, y=1..2, cond, ptions), de - ; vars ; range ; cond ; x=1..2 y=1..2 ; options .

: linecolor= ; scene=[x,y]- , ; iterations = , ( 1); stepsize=, , (x2-x1)/20, ; obsrange=true/false - , .

n - : [x0, y0, y'0, y''0,], x0- , ; y0- x0; y'0, y''0 , - , .. (n-1)- .

 

1. :

f(x)=axn, f(x)= sin(x), f(x,y)= cos(x)y3, f(x,y)= cos(x)y3,f(x)=cos(2x)2.

 

>Diff(a*x^n,x)=diff(a*x^n,x);

 

> Diff(sin(x),x)=diff(sin(x),x);

 

> f(x,y):=cos(x)*y^3;

 

> Diff(f(x,y),x)=diff(f(x,y),x);

> Diff(f(x,y),x$2,y$2)=diff(f(x,y),x$2,y$2);

 

> Diff(sin(x^2),x)=diff(sin(x^2),x);

 

> Diff(cos(2*x)^2,x$4)=diff(cos(2*x)^2,x$4);

:

 

> simplify(%);

 

> combine(%);

2. f(x)=sin2(x)/(2+sin(x)) x =p/2, x =p.

 

> y:=sin(x)^2/(2+sin(x)): d2:=diff(y,x$2):

> x:=Pi; d2y(x)=d2;

x:=p d2y(p)=1

 

> x:=Pi/2;d2y(x)=d2;

:=

3.

) , ) .

>Int(sin(x)/x,x=0..1.)=evalf(int(sin(x)/x, x=0..1),3);

 

>Int(x*exp(-x),x=0..infinity) = evalf(int(x*exp(-x), x=0..infinity),1);

4. :

) ; ) .

>Int(cos(x)*cos(2*x)*cos(3*x),x)=int(cos(x)* cos(2*x)*cos(3*x),x);

 

 

>Int((3*x^4+4)/(x^2*(x^2+1)^3),x)=int((3*x^4+4)/ (x^2*(x^2+1)^3),x);

 

5.

, a >0, b >0.

 

> assume (a>0); assume (b>0);

> Int(sin(x)*cos(x)/(a^2*cos(x)^2+b^2*sin(x)^2),

x=0..Pi/2)=int(sin(x)*cos(x)/(a^2*cos(x)^2+b^2*

sin(x)^2),x=0..Pi/2);

 

, a >-1.

> assume(a>-1);

>Int((1-exp(-a*x^2))/(x*exp(x^2)),x=0..+infinity) =int((1-exp(-a*x^2))/(x*exp(x^2)), x=0..+infinity);

 

6. y '+ y cos x =sin x cos x.

 

> restart;

> de:=diff(y(x),x)+y(x)*cos(x)=sin(x)*cos(x);

de: =

 

> dsolve(de,y(x));

1

 

7. y ''-2 y '+ y =sin x + e - x.

 

> restart;

> deq:=diff(y(x),x$2)-2*diff(y(x),x)+y(x)

=sin(x)+exp(-x);

deq:=

 

> dsolve(deq,y(x));

8. :

y (4)+ y ''=2cos x, y (0)=-2, y '(0)=1, y ''(0)=0, y '''(0)=0.

 

> de:=diff(y(x),x$4)+diff(y(x),x$2)=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)- x sin(x)+

9. :

, , . .

 

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

de: =

 

> cond:=y(0)=0,y(Pi/2)=0;

 

> dsolve({de,cond},y(x));

y(x)=2 x -p+pcos(x)

: .

 

> y1:=rhs(%):plot(y1,x=-10..20,thickness=2);

 

10. :

 

> 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) y (t), _ 1 _ 2.

 

11. :

, , ,

.

 

> restart; with(Dtools):

> DEplot(diff(y(x),x$3)+x*sqrt(abs(diff(y(x),x)))

+x^2*y(x)=0, {y(x)}, =-4..5, [[y(0)=0,D(y)(0)=1,

(D@@2)(y)(0)=1]], stepsize=.1, linecolor=black,

thickness=2);

 

 

1. f(x).

 

- -
  ) ; ) .   ) ; ) .
  ) ; ) .   ) ; ) .
  ) ; ) .   ) ; ) .
  ) ; ) .   ) ; ) .
  ) ; ) .   ) ; ) .

 

2. , .

 

   
   
   
   
   
   
   
   
   
   

 

3. . .

 

  y''(x)=x∙sin(x)   y(0)=1, y′(0)=2, y′′(0)=1 y′ = y∙z + x; z′= x2 - y2. y(0)=1, z(0)=2;
  y′′′(x)+2y′(x)=x2;   y(0)=1, y′(0)=2, y′′(0)=1 y′= y∙x + 2∙z; z′ = y - 4∙z. y(1)=1, z(1)=0.
  y′′′(x)+2y′(x)=e2x;   y(0)=1 y′(0)=2, y′′(0)=1 y′ = 4x2 + y/(x+1); z′ = e-(x2+y2) + z. y(0)=1, z(0)=0.5, a=0, b=1
  y′′′(x)+5y′(x)=e2x + sin(4x);   y(0)=1, y′(0)=2, y′′(0)=1 y′ = y∙z/x; z′ = x/z; y(1)=0, z(1)=0.5.
  10y′′(x)+7y′(x)=e7x   y(0)=1, y′(0)=2 y′ = ln(2∙x + (9x2 +z2)1/2); z′ = (9x2 +y2)1/2)+x∙sin(2∙x); y(0)=0, z(0)=0.5.
  10y′′+2y′=x∙sin(4x)   y(0)=1, y′(0)=2 y′ = z2 + x; z′ = y∙x; y(1)=0, z(1)=10.
  5y′′(x)+y′(x)=x∙e-x y(0)=1, y′(0)=2 y′ = y∙cos(x) z∙sin(x); z′ = y∙sin(x)+ z∙cos(x); y(0)=0, z(0)=2.
  3∙z′′(x)+z′(x)=x∙sin(x)   z(0)=0, z′(0)=1 y′ = 2∙cos(3∙y+z); z′ = x/(4∙y+x)+x+11; y(0)=3, z(0)=3.
  y′′′(x)+3∙y′′(x)=x2   y(0)=1, y′(0)=0 y′ = y + x∙(x+5); z′ = x - z3; y(0)=1, z(0)=-0.1.
  4∙y′′(x)+2∙y′(x)=x2   y(0)=1, y′′(0)=0 y′ = y∙z + x∙cos(x); z′ = -z2 +10 ∙x; y(0)=0, z(0)=0.2.

 



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