## 35 Reputation

1 years, 143 days

## ode system is singular at the initial po...

Maple 2021

Could you help me how to deal with this problem?

 > restart;
 > with(plots):with(plottools):with(DETools):
 >
 > Sys:=diff(T(R),R)=((1-1/R)*(sqrt(1-(alpha/R)^2*(1-1/R))))^(-1),diff(Phi(R),R)=(alpha/R)^2*(sqrt(1-(alpha/R)^2*(1-1/R)))^(-1);
 (1)
 >
 > inits:=[[T(0)=0.5,Phi(0)=0],[T(0)=0.5,Phi(0)=Pi/4]];
 (2)
 > K:=dsolve([Sys,op(op(1,inits))],[Phi(R),T(R)],numeric,parameters=[alpha],output=listprocedure);
 >

## Root finder for the fourth power polynom...

Maple 2021

Could you help me to solve this problem for the parameter beta?

 > restart;
 > e1:= 0.5; e2:=0.2;theta:=5;yeq:=e2;
 (1)
 > f:=(theta*x-1)*(1-x)*(1+beta*x^2)-y; g:=x/(1+beta*x^2); gs:=unapply(g,x);
 (2)
 > fs:=subs(y=yeq,f);
 (3)
 > assumptions:=x>1/theta, x<1,beta>0,beta<1,gs(x)>e1; solve(fs=0,x,useassumptions) assuming assumptions;
 > gs2:=subs(beta=0.6,gs(x));
 (4)
 > sol:= solve(subs(beta=0.6,fs=0),x,useassumptions) assuming x>1/theta, x<1;
 (5)
 > subs(x=sol[1],gs2);
 (6)
 >

## animate for the multiple parameters...

Maple

How can I show all the parameters in title ?

 > restart;
 > with(DEtools):with(plots):with(plottools):
 >
 >
 > sigma1:=e1*alpha: sigma2:=e2*delta:
 > g:=x/(1+beta*x^2); f:=(theta*x-1)*(1-x)*(1+beta*x^2)-y; h:=alpha*g-z-sigma1; j:=delta*y-sigma2;
 (1)
 >
 >
 > p0:=theta->plot([1/theta,y,y=0..1],linestyle=dash,color= green): p1:=e1->plot([x,e1,x=0....1.5],color=blue): q0:=animate(p0,[theta],theta=2...10): q1:=animate(p1,[e1],e1=0.1..1): q2:=plot([1,y,y=0..1],linestyle=dash,color= green): p3:=beta->plot([x,x/(1+beta*x^2),x=0..1.5],color=magenta); q3:=animate(p3,[beta],beta=0..1.5): display([q0,q1,q2,q3],view=[0..1.5,0..1]);
 >

## CharacteristicPolynimial...

Maple

I could not figure how to fix my code. Could you help me?

LinearAnalysis.mw

 > restart;
 > :with(DynamicSystems):with(DEtools):with(plots):with(plottools):with(DETools):
 > wiht(LinearAlgebra): with(VectorCalculus):
 >
 >
 > e1:=0.5: e2:=0.2:
 > sigma1:=e1*alpha: sigma2:=e2*delta:
 > g:=x/(1+beta*x^2); f:=(theta*x-1)*(1-x)*(1+beta*x^2)-y; h:=alpha*g-z-sigma1; j:=delta*y-sigma2; F:=g*f; H:=y*h; G:=z*j;
 (1)
 >
 > theta:=5;alpha:=1;delta:=1;beta:=0.4; Ffunc:=unapply(F,x,y); Hfunc:=unapply(H,x,y,z);Gfunc:=unapply(G,y,z); J1:=Jacobian([Ffunc(x,y),Hfunc(x,y,z),Gfunc(y,z)],[x,y,z]); C:= CharacteristicPolynomial(J1,lambda); R:= RouthTable(C,lambda);
 (2)
 > E0:= [0,0,0]; JE0:=subs(x=E0[1],y=E0[2],z=E0[3],J1); CE0:=CharacteristicPolynomial(JE0,lambda); RE0:= RouthTable(CE0,lambda); RouthTable(CE0,lambda,'stablecondition'=true);
 (3)
 > E1:= [1,0,0]; JE1:=subs(x=E1[1],y=E1[2],z=E1[3],J1); CE1:=CharacteristicPolynomial(JE1,lambda); RE1:= RouthTable(CE1,lambda); RouthTable(CE1,lambda,'stablecondition'=true);
 (4)
 > E2:= [1/theta,0,0]; JE2:=subs(x=E2[1],y=E2[2],z=E2[3],J1); CE2:=CharacteristicPolynomial(JE2,lambda); RE2:= RouthTable(CE2,lambda); RouthTable(CE2,lambda,'stablecondition'=true);
 (5)
 >

## Check positivity...

Maple 2021

Why did the last line is(ysol2[1]>0) give false?

Maple does not recognize the assumption?

 > restart;
 >
 >
 > interface(showassumed=0);
 (1)
 > assume(theta>1,alpha>0,sigma1>0,beta>0,sigma2>0,delta>0,x>0,y>0,z>0);
 >
 > f:=((theta*x-1)*(1-x)-y); g:=y/(1+beta*y^2); h:=(alpha*x-sigma1)*(1+beta*y^2); j:=(delta*g-sigma2); dxdt:=x*f; dydt:=g*(h-z); dzdt:=z*j;
 (2)

 > E0:=<0,0,0>;
 (3)

 > xsol1:=solve(h=0,x) assuming x>1/theta and x<1; ysol1:=solve(subs(x=xsol1,f)=0,y)assuming xsol1>1/theta and xsol1<1; E1:=;
 (4)

 > ysol:=[solve(j=0,y)]; ysol2:=simplify(subs(sigma2=delta*eta,ysol)) assuming (1-4*beta*eta^2>0 and eta>0);
 (5)
 > is(ysol2[1]>0)
 (6)
 >
 (7)
 >
 >