Differentiaaliyhtälöitä Laplace-muunnoksin

30.9.2005 HA

Alustukset

>    restart:

>    with(inttrans): alias(L=laplace,IL=invlaplace,u=Heaviside):

>   

Esim. 1

>    restart:

>    with(inttrans): alias(L=laplace,IL=invlaplace,u=Heaviside):

>    diffyht:=diff(y(t),t,t)-y(t)=t;

diffyht := diff(y(t),`$`(t,2))-y(t) = t

>    Ldy:=L(diffyht,t,s); # Maplen laplace osaa derivaattasäännön.

Ldy := s*(s*L(y(t),t,s)-y(0))-D(y)(0)-L(y(t),t,s) = 1/(s^2)

>    Ldy:=subs(y(0)=1,D(y)(0)=1,Ldy);

Ldy := s*(s*L(y(t),t,s)-1)-1-L(y(t),t,s) = 1/(s^2)

>    Y:=solve(Ldy,L(y(t),t,s));

Y := (s^3+s^2+1)/s^2/(s^2-1)

>    Y:=convert(Y,parfrac,s);

Y := -1/(s^2)+3/2/(s-1)-1/(2*(s+1))

>    y:=IL(Y,s,t);

y := -t+3/2*exp(t)-1/2*exp(-t)

>    plot([t,y],t=0..3,color=[red,blue],title="Punainen heräte, sininen vaste");

[Maple Plot]

Näin toimii vaimentamaton systeemi.

Esim. 2

>    restart:

>    with(inttrans): alias(L=laplace,IL=invlaplace,u=Heaviside):

>    diffyht:=diff(y(t),t,t)+2*diff(y(t),t)+y(t)=exp(-t);

diffyht := diff(y(t),`$`(t,2))+2*diff(y(t),t)+y(t) = exp(-t)

>    Ldy:=L(diffyht,t,s); # Maplen laplace osaa derivaattasäännön.

Ldy := s*(s*L(y(t),t,s)-y(0))-D(y)(0)+2*s*L(y(t),t,s)-2*y(0)+L(y(t),t,s) = 1/(1+s)

>    Ldy:=subs(y(0)=-1,D(y)(0)=1,Ldy); #Alkuehdot

Ldy := s*(s*L(y(t),t,s)+1)+1+2*s*L(y(t),t,s)+L(y(t),t,s) = 1/(1+s)

>    Y:=solve(Ldy,L(y(t),t,s)); factor(%);

Y := -s*(2+s)/(3*s^2+s^3+3*s+1)

-s*(2+s)/(1+s)^3

>    Y:=convert(Y,parfrac,s);

Y := -1/(1+s)+1/((1+s)^3)

>    y:=IL(Y,s,t);

y := (-1+1/2*t^2)*exp(-t)

>    plot([exp(-t),y],t=0..3,color=[red,blue],title="Punainen heräte, sininen vaste");

[Maple Plot]

>    plot([exp(-t),y],t=3..8,color=[red,blue],title="Punainen heräte, sininen vaste");

[Maple Plot]

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