> >

> > > >

> >

> >

(2)(2)

(1)(1)

> >

y:=unapply(f(x)+k*int(K(x,t)*g(t),t=0..2*Pi),x,t);restart;

G:=unapply((2*Pi-x)*int(f(t),t=0..x)+int((2*Pi-t)*f(t),t=x..2*Pi),f);lambda:=(1/(4*Pi^2)):

G := f/ 2 πKx 0

x

f t dt Cx

2 π

2 πK t f t dt

f[0]:=unapply(1-cos(x),x):f[1]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[0]))),x):f[2]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[1]))),x):f[3]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[2]))),x):f[4]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[3]))),x):f[5]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[4]))),x):f[6]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[5]))),x):f[7]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[6]))),x):f[8]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[7]))),x):f[9]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[8]))),x):f[10]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[9]))),x):f[11]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[10]))),x):f[12]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[11]))),x):f[13]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[12]))),x):f[14]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[13]))),x):f[15]:=unapply(simplify(f[0](x)+lambda*simplify(G(f[14]))),x):simplify(diff(f[1](x),[x$2])+lambda*f[1](x));

132

32 cos x π

4C4 π

2Kx2K2 cos x C2

π4

plot({diff(f[4](x),[x$2])+lambda*f[4](x),cos(x)},x);;

> >

x

K2 πK

3 π2

KπKπ2

0 π2

π 3 π2

2 π

K1

K0.5

0.5

1

plot(f[9](x),x=-10..10);]

> >

> >

(3)(3)

xK10 K5 0 5 10

0.5

1

1.5

2

2.5

de:=diff(y(x),[x$2])+lambda*y(x)=cos(x):y(x):=unapply(rhs(dsolve({de,y(2*Pi)=0,D(y)(0)=0})),x);

y x := x/

4 cos12

x

π π

2

cos 1 4 π2K1

K4 cos x π

2

4 π2K1

plot(y(x));

> >

(4)(4)

> >

K10 K5 0 5 10

0.5

1

1.5

2

2.5

restart;de:=diff(y(x),[x$2])+k^2*y(x)=cos(x):y(x):=unapply(rhs(dsolve({de,y(2*Pi)=0,D(y)(0)=0})),x);

y x := x/Kcos k x

cos 2 k π k2K1C

cos x

k2K1