TianyuCheng

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0 years, 285 days

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These are questions asked by TianyuCheng

R2,R3,R4,R5 have been given,I try to solve the following ode equation: The initial value:r2(0)=r3(0)=r4(0)=r5#(0);system={diff(r2(theta),theta)=R2;diff(r3(theta),theta)=2*R2*U2+R3;diff(r4(theta),theta)=R2*(U2^2+2*U3)+3*R3*U2+R4;diff(r5(theta),theta)=2*R2*(U2*U3+U4)+3*R3*(U2^2+U3)+4*R4*U2+R5}.I want to get the exact solution of subs(theta=2*Pi,U5),But it just gives the form include _Z1,_Z_2,_Z3.How can I get exact value?
 

restart:

#interface(echo = 2)

c10:=-26/9:c01:=-9.330480048:c11:=-199/33:c02:=-7.162134941:c12:=-93/11:c03:=-.756797589:d10:=11/9:d01:=26/9:d11:=19/3:d02:=25/3:d12:=8:d03:=1:

 

 

A1 := c03*y^3+c12*x*y^2+c02*y^2+c11*x*y+c01*y+c10*x:B1:=d03*y^3+d12*x*y^2+d02*y^2+d11*x*y+d01*y+d10*x:

aa1:=diff(arctan(v/u),u):bb1:=diff(arctan(v/u),v):

  A11:= subs({x = -u, y = (v+c10*u/sqrt(w))*sqrt(w)/c01}, -A1):B11:= subs({x = -u, y = (v+c10*u/sqrt(w))*sqrt(w)/c01}, B1):

w :=-c01*d10+c10*d01:

f11 := subs({x = -u, y = (v+c10*u/sqrt(w))*sqrt(w)/c01}, -A1): collect(%, v):f2 := subs({x = -u, y = (v+c10*u/sqrt(w))*sqrt(w)/c01}, B1): collect(%, v):g := simplify(-c10*f11/sqrt(w)+c01*f2/sqrt(w)): collect(%, v):

f111:=collect(1/sqrt(w)*f11,v):f222:=collect(1/sqrt(w)*g,v):

F1:=u*f111+v*f222:F11:=subs({u=r*cos(theta),v=r*sin(theta)},F1):F11:=F11/r:F2:=aa1*f111+bb1*f222:F22:=subs({u=r*cos(theta),v=r*sin(theta)},F2):

M:=F11/F22:M:=taylor(M,r,6):r1:=coeff(M,r,1):simplify(%):subs(c10=-d01,r1):simplify(%):

 

c10:=-d01:R2:=coeff(M,r,2):R3:=coeff(M,r,3):R4:=coeff(M,r,4):R5:=coeff(M,r,5):

 

#R2,R3,R4,R5 have been given,I try to solve the following ode equation: The initial value:r2(0)=r3(0)=r4(0)=r5#(0);system={diff(r2(theta),theta)=R2;diff(r3(theta),theta)=2*R2*U2+R3;diff(r4(theta),theta)=R2*(U2^2+2*U3)+3*R3*U2+R4;diff(r5(theta),theta)=2*R2*(U2*U3+U4)+3*R3*(U2^2+U3)+4*R4*U2+R5}.I want to get the exact solution of subs(theta=2*Pi,U5),But it just gives the form include _Z1,_Z_2,_Z3.How can I get exact value?

 

U2:=rhs(dsolve({diff(r2(theta),theta)=R2,r2(0)=0})):

U3:=rhs(dsolve({diff(r3(theta),theta)=2*R2*U2+R3,r3(0)=0})):

U4:=rhs(dsolve({diff(r4(theta),theta)=R2*(U2^2+2*U3)+3*R3*U2+R4,r4(0)=0})):

U5:=rhs(dsolve({diff(r5(theta),theta)=2*R2*(U2*U3+U4)+3*R3*(U2^2+U3)+4*R4*U2+R5,r5(0)=0})):

U6:=subs(theta=2*Pi,U5):save U6, output5:

 

 

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I try to give the plot, but it shows nothing. why? How can l find the range where 2+r*(b-2)*sqrt(b-1) can be positive, and the range where 2+r*(b-2)*sqrt(b-1) can be negative?


 

restart; with(plots, implicitplot); implicitplot(2+r*(b-2)*sqrt(b-1), b = 1 .. 100, r = 1 .. 100, scaling = constrained)

 

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Why is my figure not smooth? How can I make it smooth?
 

with(plots, implicitplot); with(plots)

pp1 := implicitplot(mu*x-ln(1+x) = 0, mu = -10 .. 5, x = -5 .. 5, color = black)

 

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s1 := RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S)

s2 := -(D1*RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S)*D6-S*D6+RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S))/(D2*D6*RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S))
algsubs(s = RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S), s2)

I try to use s1 to represent s2, and I get the following result:

-(D1*__SELECTION(RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S))*D6-S*D6+RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S))/(D2*D6*RootOf(D1*D2*D6*_Z^2+(-D1*D4*D6-D2*D6*S-D4)*_Z+D4*D6*S))
what is mean of __SELECTION, If algsubs cannot work well, what should I do?

I try to use phaseportrait to get the phase portrait of the following ode system, why it does not give the plot or the error reminding?


 

with(LinearAlgebra)

phaseportrait([diff(x(t), t) = -((y(t)^2*x(t)+x(t)^3+y(t)-x(t))*sqrt((y(t)^2+x(t)^2)/x(t)^2)-2*x(t))*x(t)/sqrt(y(t)^2+x(t)^2), diff(y(t), t) = -x(t)*((y(t)^3+y(t)*x(t)^2-y(t)-x(t))*sqrt((y(t)^2+x(t)^2)/x(t)^2)-2*y(t))/sqrt(y(t)^2+x(t)^2)], [x(t), y(t)], t = 0 .. 50, [[x(0) = -1, y(0) = -1], [x(0) = -5, y(0) = -5]], x = -10 .. 20, y = -10 .. 30, dirgrid = [40, 40], stepsize = 0.1e-2, numframes = 90, axes = BOXED, linecolor = black)

phaseportrait([diff(x(t), t) = -((y(t)^2*x(t)+x(t)^3+y(t)-x(t))*((y(t)^2+x(t)^2)/x(t)^2)^(1/2)-2*x(t))*x(t)/(y(t)^2+x(t)^2)^(1/2), diff(y(t), t) = -x(t)*((y(t)^3+y(t)*x(t)^2-y(t)-x(t))*((y(t)^2+x(t)^2)/x(t)^2)^(1/2)-2*y(t))/(y(t)^2+x(t)^2)^(1/2)], [x(t), y(t)], t = 0 .. 50, [[x(0) = -1, y(0) = -1], [x(0) = -5, y(0) = -5]], x = -10 .. 20, y = -10 .. 30, dirgrid = [40, 40], stepsize = 0.1e-2, numframes = 90, axes = BOXED, linecolor = black)

(1)

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