<Text-field style="_cstyle11" layout="_pstyle10">Figur 1</Text-field>

Figurerne til noterne Differentialligninger af f\303\270rste orden

Preben Alsholm, 14/8 2007

restart;

with(DEtools): with(plots):

ligning:=diff(x(t),t)=t^2+x(t)^2;

DEplot(ligning,x(t),t=-1..1,[[x(0)=-0.4]],x=-1..1,stepsize=.1,color=blue,linecolor=red);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 2 Eksempel 18</Text-field>

ligning:=diff(x(t),t)=(1-x(t))^2*2*t;

dsolve(ligning);

dsolve({ligning,x(0)=1/2});

DEplot(ligning,x(t),t=-3..3,[seq([x(0)=k/2],k=-1..2),seq([x(0)=k/4],k=5..7)],x=-1/2..2,stepsize=.1,linecolor=red, color=blue);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 3 Eksempel 20</Text-field>

ligning:=diff(x(t),t)=1/(t^2+1)*x(t)*(1-x(t));

dsolve(ligning);

dsolve({ligning,x(0)=2});

DEplot(ligning,x(t),t=-5..5,[[x(0)=2],[x(0)=1], [x(0)=1/2],[x(0)=0],[x(0)=-1/2],[x(0)=exp(Pi/2)/(exp(Pi/2)-1)]],x=-2..4,stepsize=.1,color=blue,linecolor=red);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 4 Eksempel 21</Text-field>

ligning:=diff(x(t),t)=3*x(t)^(2/3);

dsolve(ligning);

f:=piecewise(t<k1,(t-k1)^3,t<k2,0,(t-k2)^3);

plot([subs(k1=-1,k2=2,f),subs(k1=0,k2=0,f)],t=-2..3,-1..1,thickness=2,color=red,linestyle=[0,3]);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 5 Eksempel 24</Text-field>

ligning:=diff(x(t),t)+2*x(t)=5*sin(t);

dsolve(ligning);

DEplot(ligning,x(t),t=-1..5,[seq([x(0)=k],k=-2..2)],x=-5..5,stepsize=.1,color=blue,linecolor=red);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 6 Eksempel 26</Text-field>

ligning:=t*diff(x(t),t)+2*x(t)=t*exp(-t);

res:=dsolve(ligning);

DEplot(ligning,x(t),t=0.01..1,[seq([x(1)=2-5*exp(-1)+k/200],k=-2..2)],x=-1/2..1/2,stepsize=.01,color=blue,linecolor=red);

subs(_C1=2,rhs(res));

limit( %, t=0);

<Text-field style="_cstyle11" layout="_pstyle10">Figur 7 Eksempel 27</Text-field>

ligning:=diff(x(t),t)=t^2+x(t)^2;

dsolve({ligning,x(0)=1});

f:=unapply(rhs(%),t):

Tabel:

Matrix([[t,seq(.05*k,k=0..6)],[x,seq(evalf(f(.05*k)),k=0..6)]]);

L1:=dsolve({ligning,x(0)=1},x(t),type=numeric,output=listprocedure,method=classical,stepsize=0.1);

eul1:=subs(L1,x(t));

Matrix([[t,seq(.1*k,k=0..3)],[x,seq(evalf(eul1(.1*k)),k=0..3)]]);

L2:=dsolve({ligning,x(0)=1},x(t),type=numeric,output=listprocedure,method=classical,stepsize=0.05);

eul2:=subs(L2,x(t));

Matrix([[t,seq(.05*k,k=0..6)],[x,seq(evalf(eul2(.05*k)),k=0..6)]]);

plot([f(t),eul2(t),eul1(t)],t=0..0.5,color=red,linestyle=[0,2,3],thickness=[4,2,2],legend=["Eksakt","Euler med h = 0.05","Euler med h = 0.1"]);