> # Teht. 2 # # > vp:=diff(y(t),t,t)+2*y(t); / 2 \ | d | vp := |----- y(t)| + 2 y(t) | 2 | \ dt / -------------------------------------------------------------------------------- > # Kehitetään oikea puoli (r(t)) sinisarjaksi, josta otetaan "astelukuun" 7 saakka. # > bn:=(2/Pi)*int(f(x)*sin(n*x),x=0..Pi); Pi / | | f(x) sin(n x) dx | / 0 bn := 2 --------------------- Pi -------------------------------------------------------------------------------- > bn:=simplify((2/Pi)*int(sin(n*x),x=0..Pi)); cos(n Pi) - 1 bn := - 2 ------------- n Pi -------------------------------------------------------------------------------- > bn:=subs(cos(n*Pi)=(-1)^n,bn); n (-1) - 1 bn := - 2 --------- n Pi -------------------------------------------------------------------------------- > s7:=sum(bn*sin(n*x),n=1..7); sin(x) sin(3 x) sin(5 x) sin(7 x) s7 := 4 ------ + 4/3 -------- + 4/5 -------- + 4/7 -------- Pi Pi Pi Pi -------------------------------------------------------------------------------- > n:='n':rjono:=seq(bn*sin(n*t),n=1..7); sin(t) sin(3 t) sin(5 t) sin(7 t) rjono := 4 ------, 0, 4/3 --------, 0, 4/5 --------, 0, 4/7 -------- Pi Pi Pi Pi -------------------------------------------------------------------------------- > dy1:=vp=rjono[1];dy2:=vp=rjono[3];dy3:=vp=rjono[5];dy4:=vp=rjono[7]; / 2 \ | d | sin(t) dy1 := |----- y(t)| + 2 y(t) = 4 ------ | 2 | Pi \ dt / / 2 \ | d | sin(3 t) dy2 := |----- y(t)| + 2 y(t) = 4/3 -------- | 2 | Pi \ dt / / 2 \ | d | sin(5 t) dy3 := |----- y(t)| + 2 y(t) = 4/5 -------- | 2 | Pi \ dt / / 2 \ | d | sin(7 t) dy4 := |----- y(t)| + 2 y(t) = 4/7 -------- | 2 | Pi \ dt / -------------------------------------------------------------------------------- > yr1:=a1*cos(t)+b1*sin(t):eval(subs(y(t)=yr1,dy1));\ sin(t) a1 cos(t) + b1 sin(t) = 4 ------ Pi -------------------------------------------------------------------------------- > a1:=0;b1:=4/Pi; a1 := 0 4 b1 := ---- Pi -------------------------------------------------------------------------------- > yr2:=a2*cos(t)+b2*sin(t):eval(subs(y(t)=yr2,dy2)); sin(3 t) a2 cos(t) + b2 sin(t) = 4/3 -------- Pi -------------------------------------------------------------------------------- > a2:=0:b2:=4/(3*Pi); 4 b2 := ---- 3 Pi -------------------------------------------------------------------------------- > yr3:=a3*cos(t)+b3*sin(t):eval(subs(y(t)=yr3,dy3)); sin(5 t) a3 cos(t) + b3 sin(t) = 4/5 -------- Pi -------------------------------------------------------------------------------- > a3:=0:b3:=4/(5*Pi); 4 b3 := ---- 5 Pi -------------------------------------------------------------------------------- > yr4:=a4*cos(t)+b4*sin(t):eval(subs(y(t)=yr4,dy4)); sin(7 t) a4 cos(t) + b4 sin(t) = 4/7 -------- Pi -------------------------------------------------------------------------------- > a4:=0:b4:=4/(7*Pi); 4 b4 := ---- 7 Pi -------------------------------------------------------------------------------- > yr1,yr2,yr3,yr4; sin(t) sin(t) sin(t) sin(t) 4 ------, 4/3 ------, 4/5 ------, 4/7 ------ Pi Pi Pi Pi -------------------------------------------------------------------------------- > hy:=vp=0; / 2 \ | d | hy := |----- y(t)| + 2 y(t) = 0 | 2 | \ dt / > yh:=dsolve(hy,y(t)); 1/2 1/2 yh := y(t) = _C1 cos(2 t) + _C2 sin(2 t) -------------------------------------------------------------------------------- > ylratk:=rhs(yh)+yr1+yr2+yr3+yr4; 1/2 1/2 704 sin(t) ylratk := _C1 cos(2 t) + _C2 sin(2 t) + --- ------ 105 Pi -------------------------------------------------------------------------------- > dylratk:=diff(ylratk,t); 1/2 1/2 1/2 1/2 704 cos(t) dylratk := - _C1 sin(2 t) 2 + _C2 cos(2 t) 2 + --- ------ 105 Pi -------------------------------------------------------------------------------- > aey:=eval(subs(t=0,ylratk))=0,eval(subs(t=0,dylratk))=0; 1/2 704 aey := _C1 = 0, _C2 2 + ------ = 0 105 Pi -------------------------------------------------------------------------------- > c:=solve({aey},{_C1,_C2}); 1/2 352 2 c := {_C1 = 0, _C2 = - --- ----} 105 Pi -------------------------------------------------------------------------------- > ratk:=subs(c,ylratk); 1/2 1/2 352 2 sin(2 t) 704 sin(t) ratk := - --- ---------------- + --- ------ 105 Pi 105 Pi -------------------------------------------------------------------------------- > plot({subs(x=t,s7),ratk},t=-Pi..Pi,title=`teht 2`); -------------------------------------------------------------------------------- > s7; sin(x) sin(3 x) sin(5 x) sin(7 x) 4 ------ + 4/3 -------- + 4/5 -------- + 4/7 -------- Pi Pi Pi Pi -------------------------------------------------------------------------------- > plot(eval(sign(t)),t=-1..1); -------------------------------------------------------------------------------- > sign(-1);\ -1 -------------------------------------------------------------------------------- >